Magic is treated as a resource that can do anything, but it is in limited supply. How would it affect the world and technological advancement? Will they develop phones when telepathy is available? Will they require matches if they could just cast a spell?
In the words of Sir Arthur C. Clarke...
Any sufficiently advanced technology is indistinguishable from magic. — Clarke's Third Law
Bring a person from 2000 years back to us here today. What would they find?
No person can ever again get afflicted by smallpox. The warding sign that apparently shields us from this deadly malady is a curious stylized sun-shaped mark on the shoulder or arm. Magic!
A person points a curious smooth metal box with an eye at them, and in an instant it has captured an exact replica of them. Not only that but this little box can also imitate what they did and even said, capturing their motions even and their voice. Magic!!
A person can float hundreds of leagues above the Earth, and not fall to the ground. In fact, people have even walked on the Moon... and brought pieces if it back to us, that we can see and touch. Magic!!!
Scurvy is no longer a sickness. Just eat this curious orange fruit from China every once in a while and you will not get ill and die from Scurvy. MAGIC!!!
The point is that pretty much everything we do in the modern world is magic to anyone that lived just a couple of thousands year back. You want to have a magical resource that can do almost anything but is not infinitely available? Electricity! Pure magic, when you do not know what it is.
So to slightly rephrase your question: how would the invention of some hitherto unknown technology change the world?
As you can see this question is much too broad to be answered. Anything and everything can happen. You as an author are free to make anything you like up. Just treat this "magic" as you would any kind of invention or application of a scientific discovery.
This is broad, so I will answer it in a general way.
The main limiting factor here is: rarity. Humans like to do what they do faster, cheaper, simpler, and better, whatever that may be. The answer to this question really depends on how rare magic is and what the COSTS are. If only a few people can work magic or the resources to cast are dear, then humans will come up with a better alternative, so you may, indeed still have tech, depending on that cost. A good example is fossil fuels. We actually do have alternative methods of collecting energy, but until the cost of fossil fuels is GREATER than the cost of building and maintaining those alt methods, we simply won't use them very much.
What's the advantage?
If magic is rare but present, there should be an advantage to using magic. It could be a tool of elite members of their society. Magic, at high levels, could be the equivalent of a cell phone in the late 1980s--something that only certain people use and is very expensive (it was about 1/4 of the average income to buy at one point).
There may be more tech, sooner, as a result I can see a smart person in a city that doesn't have the money to afford a mage coming up with systems to do the equivalent with tech. In fact, in I can see common, magicless folk deciding that they want to achieve what magic can, without magic. Before a technology is a technology, it's an idea, and it has to be something that people can easily conceive of before it can be developed. And if magic can do what modern tech can do, you may have a lot more people trying to develop it, more quickly, and more often than in the past.
This early development may completely change the face of tech For about a 1/2 minute, there was something called a steam-driven car, which was phased out because of the standard internal combustion engine we know and love today. But, if magical horseless carriages are a thing (albeit a rare one) in your world, they might develop earlier and have more staying power as a trend. This is one example but yes, I can see steam power coming sooner and being more ubiquitous. There are always limits on tech because of manufacturing ability, but it's possible that too will be pushed as people figure out what they need to make in order to have what magic gives. I think the same might happen with hot air balloons. The first manned flight happened in the 1780s but hey, it might happen sooner, if there's a push from flying, magical warfare. Gosh. This is getting really steampunky...
EDIT:Basically, look at outmoded tech, such as pneumatic tubes, which was phased out because of other tech, but that, if developed and used earlier, may become more widespread because it won't be replaced as quickly.
Everything Erin said.
However, I would like to emphasize some things:
- If Magic is rare but better, only rich people will have magic.
- If magic is rare but much better, only people with magic will be rich.
- If magic is rare but as good as tech, you would only really hear about it on shows like CSI.
- If magic is rare but worse then tech, only poor people would have it (flaky telepathy would be the poor man's iPhone).
You could look at the Dresden Files for a similar setup. In that world, magic exists, but anyone who uses magic or is a wizard is doomed to blow technology up the moment they get near it. Can't drive cars, use cell phones, flashlights, etc.
I think "magic" can be separated from technology and be plausible. Shooting fireballs and summoning demons doesn't really translate into usable technological advancements. Could be turned into weapons by the military and governments across the world of course, and that could even be a plot point in a story, but it can still be gimmicked to function within you "normal" world of regular advancements.
The real challenge would be to use said gimmicks to your advantage in someway to make it seem like magic exists in a modern world (if that's what your going for) without contributing much to scientific advancement. Or, you could go the other direction and make wildly innovative techs that DID use magic for their advancements.
I think that if magic did really exist, it would be in our world much like how it is in the Dresden files. Relatively unknown, widely disbelieved, and basically used by a secret society of those "in the know" who have a penchant for its use.