It might be helpful to frame this question in terms of how your
world's secondary schools might teach several subjects differently
from our own:
We've all heard Clarke's maxim that "any sufficiently advanced technology is indistinguishable from magic," so how would people in a merged world teach magic? If it's a fundamental law of nature, why wouldn't it be taught in science class? How would it be taught? It's not pseudoscience if it's real. Would "divination" be included in the scientific method?
If one of the magical abilities present in your world is the ability to read or control minds, then the nature of history becomes difficult. If the Archvizier of Dunlop suddenly became a murderous despot in 1277, who's to say that she wasn't brainwashed to aid her cousin's coup later that year? Maybe her cousin wasn't the saint she's always been portrayed as. Think of all the crazy crackpot historical theories in our world - Bigfoot, the Chupacabra, the Illuminati - and realize that in your world, they're all possible.
On the face of it, this would seem to be the same in your merged world as it is in ours, but if a divination spell can instantly prove any theorem or solve any algorithm, why would complex mathematics such as calculus have ever emerged? However, if there is an "arithmancy" that governs magical work, the line between science and magic begins to blur and the two might aid one another.
As other answers have noted, the economics of such a world would be vastly different than our own. If something is cheaper when done magically, the scientific supply of that good or service will dry up. However, if science does something cheaper than the magical equivalent, what happens to the person who can perform that service inherently? For instance, if scientific transportation is cheaper than magical teleportation, what happens to the people who have the magical ability to teleport others through space? Do they ply their trade to the wealthy, or do they try to earn money by other means?
"ilinamorato, that's silly," you say, "this question is ABOUT technology!" But hear me out. Cheap conjuration of basic objects would mean that mass manufacturing never developed, so other forces would have to have driven scientific development. We might see a much earlier development of digital information systems. Conversely, if magical conjuration could be applied to the manufacturing process, you might find yourself in a world with far cheaper, more quality-built manufactured objects.
In such a world, science fiction and fantasy would not be a pseudo-subcultural genre, but a genre that encompassed their everyday lives. So what would "genre fiction" be like? Do nerds read books about life in a magicless world - our world?
If the magic in your world allows for quick travel to distant lands, foreign language will be a much more important class to take. If it allows for time travel, the dead languages of the past and uninvented languages of the future must be considered. You wouldn't want to offend your future self.
Don't think that just because magical spells can heal injuries and diseases that health won't be a concern. Think about the processes that will be accepted in this world that might be considered homeopathy or witch doctoring in our world. Medicine will likely be a partnership between the magical and scientific, with much of the diagnostic work in the scientific realm while treatments are conducted by the mage. How will their interaction be? Will it be like in the TV show Scrubs, where surgeons and internal medicine doctors are often at odds? Will hospitals include altars for sacrifices and incantations?
Do humans integrate technology into their bodies? Do they try to select for magical attributes in their offspring? Do magical creatures such as griffins and dragons exist in this world? Is there a demihuman race such as dwarves in this world?
Does everyone integrate both magic and technology into their lives, or are there "technologists" and "magicists", neither of whom trust the other? Is technology considered "old fashioned," used only by hipsters and grandparents who are hopelessly out of touch with the new generation?
Politics and Government
How will the politics of this blended world work themselves out? Are the mages governed by a different body than the scientists? How does a government prevent magical tampering with elections? Are officials even elected, or does a diviner simply cast spells to discover who the people want to be elected, or who would be better to rule? How do different nations interact with one another? Is magic more prevalent in one country than another? Can magical power make a country a superpower?
Is there a sport that consumes the world (like football, or maybe football)? Is it scientific or magical? Is it considered poor sportsmanship to use magic in sports, or is it considered "just part of the game"?
Art and Music
This probably wouldn't be much different in the blended world than in ours, but it is worth thinking about how the arts in a magical world would look.
Clearly a magical criminal could do a lot of damage. But so could a technologically-advanced criminal. How does the police in this world fight crime? Is it an alliance between mages and techies, or is there a Magic Police and a Science Police? How does one report an emergency situation? In what manner do the magical police arrive?
Some short thoughts:
Meteorology: Will controlling the weather be as much the domain of meteorologists as predicting it?
Astronomy: Is Astrology part of this science?
Philosophy and Ethics: What is right and wrong in a world where anything is possible?
Agriculture: How is food created/grown?
Communications: How are messages transmitted?
Obviously you don't have to have an answer for every question. But as you're developing your world, it would be helpful for you to be
able to answer most of the big ones, so that when the little ones come
up you can make a decision based in the lore you've already