Why navigate the Atlantic?
The reason was economic. Before the 1200's, almost anything in the Asia's far East (i.e. after Persia) was a complete mystery to Europeans. There were just a very few vague, scarce and sparse informations, rumors and legends. That, of course, until Marco Polo actually travelled to there* departing from Italy, getting back alive (24 years later - it was a looooooong journey), and being able to tell his memories to a friend who decided to write'em down in a book and publish it.
* - Well, he claimed that he went there, but there is some doubt and debate about that being or not true.
Marco Polo described a rich Empire and culture in the far, far away East, called Cathay (which is China). He also described what he knew about central Asia, India, Mongolia, Cipango (Japan) and Burma (present day Myanmar). Goods were already traded indirectly between Europe and China, slowly passing through a long chain of intermediary merchants from many different countries, cultures and languages, such as byzantines, turks, berbers, egyptians, arabians, armenians, parthisans, persians, indians, tibetans, mongolians and chineses. Of course, with that such long chain, any goods coming from India or China had a very high price when it eventually reached Europe some years later, because all of the intermediaries would profit from it.
Merchants in the Europe, knew that the goods coming from China and India (specially China's silk and India's spice) accumulated a very high price in the way, so if someone could somehow go there and bring back goods without relying in so many intermediaries, they could sell them at lower prices and higher profits.
With the fall of Constantinople, in 1453, the trade got harder. This happened because the Muslim cultures that dominated Asia's West, Central Asia and North Africa were very hostile to the Christian cultures in Europe (many of them, catholic crusaders). With the fall of Constantinople to the Muslim Ottomans, the situation just got worse, since the last remaining land route between Europe and Asia's far East was closed, without any sea route being known. The result was that India's and China's goods which already were rare and expensive in Europe, became still more rare and expensive.
So, Europeans had very good reasons to try to reach India and China through the seas. Anybody who could manage to do that, would open a highly profitable trade route with no, or perhaps just a very few, intermediaries.
Which countries sailed or not sailed and why?
Opening new sea routes in the Atlantic to the far East would be very expensive on itself and demand a level of organization and financing that only kings could have. Which states could provide that?
The Holy Roman Empire was a mess, with all of his internal states struggling and warring under a puppet Emperor which actually lacked power.
Italy was the source of a lot of skilled sailors, but was a mix of small disputing states and city-states, so no state could finance such thing.
The Pope already had enough problems to worry about, including Muslims and heretic Christians.
East European states lacked enough organization and navigation skills, and some of them already had problems with the Ottomans.
England and France were warring during most of the XV century, wars which included Flanders, Burgundy and Austria, so all of them had too many domestic problems to worry about. Also, they already had fairly strong economies, so investing resources in navigating the Atlantic to reach India would be a silly way to waste money that they could instead invest in something else much more economically plausible.
Denmark, scandinavian states and Iceland were just too far away in the North to have any interest in navigating warmer waters.
The Muslims were not interested in sea trade through the Atlantic, because they already effectively traded by land or by navigating the Mediterrenean Sea or the Indic Ocean, and they had no interest in trading with European northern states. So, navigating the Atlantic would be pointless for them.
Far East cultures had no reasons to try to navigate the seas to reach Europe. It was damn too far away, and they already had many people to trade nearby either by land or sea. Also they knew almost nothing about Europe, and most of what they actually knew, was obtained from Persian and Arabic sources, which depicted Christian Europe very negatively.
Sub-saharian cultures were far too primitive and underdeveloped to think anything about navigating the Atlantic. The american indians also werer far too primitive and had no reason to cross the Atlantic in the other way and reach Africa or Europe.
This leaves only two suitable countries: Portugal and Spain.
During the XV century, Portugal and Spain fighted the muslim kingdom of Granada during the Reconquista war. Portugal finished its part of the war first, so they got a head start. With the war end, Portugal needed some sort of economic activity which would ensure high profit for them, and they already were skilled sailors.
Portugal started to explore Africa's coast in 1418. Their sailing skills leaded to the invention of the caravel, a type of seaship who could sail further and faster than the other existent seaships at the time and could even sail against the wind.
Spain was decided to not be kept behind Portugal, so they started to quickly develop their navigation. However, they were still struggling in war against the moorish kingdom Granada.
Crossing the Atlantic
And then, in 1486, a man called Cristoforo Colombo met the Spanish king and the Spanish queen with a crazy lunatic idea: he wanted to reach Indies by sailing west around the globe instead of sailing around Africa.
The Atlantic Ocean was a formidable obstacle back in the XV century. People already knew that the world was a spheric body, but Columbus believed that the distance between the Canary Islands to the Japan, by sailing west, should be roughly 3700 km, but in fact it is 12500 km - i.e. he severely underestimated the size of the globe. Also, he tought that Japan should be nearer to the equator and be as large as China.
Most "main-stream" navigators would not venture in such voyage because they (rightly) tought that Earth was much larger than that, so that such voyage would be a crazy idea - and they would be right if there was no American Continent in the way and the Atlantic and Pacific Oceans just formed a very large ocean.
Columbus already talked to Portugal's king in 1485, but his idea were rejected as being unrealistic, since the king's experts asserted that the Earth should be much larger than what Colombo calculated (and it is).
Spain also initially rejected Columbus's idea in 1486, but the king decided to keep he around to avoid that he went to tell his idea to somebody else. Also, in 1488, Portugal reached the Cape of Good Hope and was definitely not interested anymore in trying to sail west. Portugal was dedicated into the task of reaching India by circumnavigating Africa and nobody else was seriously competing with them yet.
In January of 1492, Spain could finally dismiss Granada once and for all, and then the king could finally dedicate all the efforts into navigation. Columbus continually kept lobbying the Spanish court for those years, and when he was just finally giving up and leaving, in April, the spanish king decided to give Columbus' idea a try. It was really a very far-fetched crazy idea, but if he was correct, Spain would quickly profit and be able to compete with Portugal, and since Spain's king knew that Columbus was a very skilled sailor, this could not be as crazy as it seemed to be afterall.
So, only seven months after expelling the Moors, Spain sent Columbus in his sea journey to the west. He departed from Spain in August, and from the Canary Islands (the last then-know piece of land in his journey), in September. After five weeks sailing unknown waters to the west, he landed in the other side of the ocean visiting islands that today are parts of Bahamas, Cuba and Haiti.
Portugal reached India in 1498, but landed in South America in 1500 and also started to explore the american continent. France, England and Flanders (Netherlands) joined the race somewhat late to the party.
Ok, how to make that not happen?
There are many points in the history around the last years of the XV century that if just a few things were different, the american continent would be only discovered much later, possibly well into the 1700's, possibly even after Australia's discovery. In fact, its discovery is much more a product of random luck and fortuitous economic situation than anything else.
If Portugal reached India before Columbus reached America, it is possible that no one (including Spain monarchs) would seriously believe the possibility of navigating to the west far away enough to reach land for some long time. Specially since they already knew at the time that Earth was too large for that being viable. Putting all the efforts into navigating around Africa would be economically safer.
What would happen if Columbus' caravel were hit by a hurricane and perished to never be seen again? Or if they all were just killed by american indians?
What if Spain monarchs just dismissed Columbus the same way as Portugal did?
What if Columbus never born to start with?
What if Portugal and Spain had some other economic interest (and perhaps didn't invented the caravel afterall)? Or if Granada resisted harder? Or if the Spanish monarchs just suddenly died in an accident and a sucession dispute began?
What if the Ottomans decided to earn gold by seriously trading with some European nations instead of keeping ressenting anti-crusader feelings?
What if Egypt decided to earn gold by routing India's goods from the Red Sea to the Mediterrenean Sea, making both the ideas of sailing west the Atlantic or around Africa too costly, too dangerous and too time-consuming?
What if Marco Polo never came back to Europe?
What if Constantinople resisted longer?
With that, the reason to not cross the Atlantic would be just: "Because it is too dangerous, too expensive and is very unlikely to be worth anything. Further, nobody ever heard of anything beyond those seas and have no reason to believe that India or Japan are anywhere near in that direction, so traveling that way is absolutely pointless."
Note that those changes are very simple, plausible and mundane, so there is no need to have supersized shipworms, nor dragons or sensational creatures, nor crazy climatic phenomena, nor crazy geographic or geologic features, nor strange geomagnetic or astronomic phenomena, nor specially unfavourable and unluckily positioned wind, sea-currents patterns or reefs, nor anything severely different than what history was until that point.
What about crossing the Pacific instead?
As soon as the news about crossing the Atlantic and what was found there spread into Europe, cartographers started to debate about what they actually were, and concluded correctly in the first years of the XVI century that those new lands could not be by any means parts of Asia, and they were in fact an entire new continent.
What made the Pacific be eventually crossed is the fact that the discovery of the Americas sparked the interest to actually circumnavigate the globe and more importantly, perhaps discover some other unknown continents or profitable trade routes.
The first circumnavigation of the globe was completed in 1522, after a long and dramatic 3-years journey which killed most of the crew and left some of them in the way, with just a few survivors with bad health being able to complete it (18 out of 237). Only 30 years before (in 1492), the idea of sailing west the Atlantic were considered lunatic, but now the globe could be circumnavigated completed exactly in that way.
If nobody crossed the Atlantic, crossing the Pacific sailing westward would have no sense. Crossing it from Asia eastward, was already pointless: "There is nothing valuable out there, just a few small and sparse worthless islands inhabited by primitive people". So even if a few people from polynesian islands actually reached the american continent and could come back to their homes, they probably won't be able to spread the news efficiently, and in the best possibility, it eventually would be just another strange fantasy legend in the folklore of a primitive tribe from a remote island that nobody in the civilized world would care to hear.
So, with the discovery of the Americas, sailors started to fearless navigate into the open ocean instead of just navigating around known coasts. With that, many remote islands around the world were discovered and charted, and it is unlikely that they would be anytime soon if the discovery of the Americas was further delayed.