The answer to your question is very long and complicated, but the long answer is probably not. There are a lot of reasons why, but some of the most prominent are...
TL;DR: Countries either get ahead through an abundance of resources and manpower, or profitable trade and cross-pollination of ideas when countries lack those resources. Your two isolated countries have none of these things. By definition due to its smaller size it's going to have less resources and population, and if it's isolated it's going to have no way to keep up with other countries by controlling trade routes or importing foreign advancements in technology.
- Resources. Smaller countries tend to have less resources. If they're on an isolated island those resources are going to be even less. If they're isolated they can't easily trade for resources they're missing. Missing some of those resources could be potentially crippling in terms of technological development.
For example the two main ingredients of bronze: copper and tin, are almost never found together, and bronze manufacture typically involves extensive cross-continental trade. You can make bronze using arsenic and copper, but the fumes are toxic and it's not as good. What happens if your island has no copper deposits? You're much less likely to ever realize the usefulness of bronze, and you'll be much less likely to develop iron working because iron working involves much hotter temperatures and is harder to smelt than copper. What happens if your country has no easily accessible deposits of coal or oil? We already see today how access to fossil fuels can make or break a country's economy. Gunpowder might be easy to manufacture from seabird guano, though.
As an example, it's been suggested that one reason why China never tried to become as much of an expansionistic empire as Rome or the later European colonial empires or never had much of an interest in exploration is because their location in east-central Asia gave them easy access to pretty much everything they could ever want. They had easy access to iron, copper, spices, silk, etc., in contrast to Europeans who had to go halfway around the world to get access to spices. They did make expeditions and territorial expansions to acquire better horses from Ferghana, control the Tarim Basin, and expand south to acquire bananas and more directly acquire spices, but they weren't dependent on trade.
- How isolated is "isolated"? Humans have a tendency to get everywhere. Our species had colonized every major landmass by about 700 CE, and did so with what amounts to Stone and Iron Age technology (the Inuit, Norse, and the numerous Polynesian cultures were not using modern technology to find these landmasses). Even really isolated landmasses often have a history of multiple waves of colonization (e.g., the native Malagasy seem to be descended from immigrants from both Africa and southeastern Asia).
The closest to what you want is the Sakoku period of Japan, where Japan closed its borders for 200 years. However, even during the Sakoku period Japan wasn't 100% isolated, it still maintained trading relationships with the Dutch for example, even if the Dutch were limited to two cities. Additionally, it wasn't like the rest of the world didn't know that Japan existed. On top of that, part of the reason sakoku ended was because Japan realized the rest of the world was advancing much faster than it, and it had the choice of either modernizing or falling behind in the technological arms race and get stomped on when someone did decide to invade them. Hermit kingdoms never do well. That said Japan does show it is possible for an island nation to remain isolated for 200 years and still have technology, but Japan had extensive contact with the rest of the world beforehand and definitely did not keep up with the rest of the world until sakoku ended.
A smaller country is also going to do more poorly if an enemy can circumvent its natural defenses of being isolated. Being island nations has helped a lot to reduce invasions for many countries, but a small island nation by definition is going to have a harder time levying a comparable army to a large island nation. What's going to decide any military campaign is if that larger nation can ship the army over and bring their greater military strength to bear.
- No large, domesticatable animals. Islands as a rule tend to have fewer large animal species than the mainland due to fewer chances for dispersal. The smaller land area means that fewer large herbivores can be supported, which means fewer carnivores. Additionally, when humans come to an island they tend to slaughter everything bigger than a cat on it, often by sheer accident. It's not hard because these large island animals tend to have small population size and aren't familiar with humans. Examples include the megafauna of Cuba, Hispaniola, Puerto Rico, Hawaii, New Zealand, New Caledonia, Madagascar, Cyprus, Sicily, the giant tortoises of the Bahamas, and so on. Even the British Isles show signs of extinction. There is no evidence that humans have ever domesticated island megafauna despite the fact that they should be highly amenable to domestication due to their passivity. The Balearic goat Myotragus has been suggested to have been tamed by the natives of that island, but this is disputed.
As a result, for the most part any domesticated animals your society has are going to be ones they brought with them. Dogs are highly likely, as are chickens and maybe rats (see: Polynesians). Large ungulates are going to be harder to bring across, especially cows and horses.
This is going to have impacts on your society. No domesticated megafauna means fewer easily available sources of meat and possibly milk or eggs. Large domesticated animals are helpful in many ways. They make the usefulness of wheels much more self-evident. They make it easier to transport goods and information. They make large-scale construction easier. They make for good food sources because you get a lot of food for little unit effort. In the American West bison and later cattle were considered an easy food source for workers because they had a lot of meat per animal.
There's also the issue that even if your society does have these things, it will also be harder to recover from fluctuations in natural resources. If the forests of your island are cut down you aren't getting any more wood. If the horses of your island die off from a plague they aren't going to be coming back from another area. You're stuck with the limited resources you have. Example: Rapa Nui. Once the forests of Rapa Nui were cut down, they didn't come back. This is very important is you have a 1700's level society, you're probably going to have rampant deforestation somewhere, whether it is for building cities, cleared for farmland, or cut down for fuel or boats.
- Reduced rates of scientific and technological advancement. Your island civilization is going to have slower rates of technological development. Full stop. No ifs, ands, or buts. There is a very simple reason for this: you have fewer brains at any one time thinking about solutions to problems, and you have no cross-pollination with other cultures.
Here's an example. Lets say you have an problem with poor farming tools. In a small country you have 50,000 people trying to solve this issue every day, and on the mainland you have 500,000 people dealing with this problem every day, plus by proxy the millions that country is connected to due to trade routes. The mainland is going to solve the problem faster because it has more people working on it.
This can be seen today. Part of the reason science is advancing at such a fast pace today is because 1) there have never been more humans alive than there are right now, and 2) those humans are highly connected and can easily share information. Isolated discoveries are much more likely to be lost or forgotten, and rapidly spread everywhere.
This is also why civilizations positioned at the center of trade routes tended to be so prosperous. Not only do they get fantastically wealthy due to all the trade routes coming through them, but they also get all sorts of new ideas coming through them. This is one reason why the Middle East, and in fact during the Islamic Golden Age the caliphates leveraged this to make cities like Bagdad a center of learning.
Additional Story-Telling Things
From a story-telling perspective, the issue you are going to be facing is literally every aspect of your society's culture is going to be different from the mainlanders. Their language and writing will be completely different. Their math system is liable to be different. Their calendar will likely be different. Other historical societies often have some differences, but here the differences will be exaggerated because every single advancement between your two island countries and the rest of the world will have been made independently, and different people will come up with different terms and ideas to explain the same things.
On top of that, the historical "lessons" that their societies have learned will be completely different. A lot of human history has been influenced by what a society has learned through previous events. Modern society has become fanatically anti-genocide and anti-eugenics after we saw where those ideas led after World War II, even though before that such behavior was condoned or even encourages (e.g., eugenics movements in the 20s). Historical Chinese dynasty's rampant misogyny was in part codified by the tyrannical reign of Dowager Empress Lü Zhi, and even centuries after her death she was used as an example by the reigning elite as to why women couldn't be trusted with power. Your societies are going to have very different ideas as to what is "right" or "wrong" based on their completely separated histories.
