Characters are smarter: Nearly every character in HPMOR has received a marked intelligence boost from their canonical versions. (The obvious exception is Hermione; she hasn't actually been boosted for multiple reasons, but she often feels that way to new readers because she's allowed to use her intelligence for far more than expositing facts from books and learning spells quickly.) Since all the characters have been that way their whole lives, this results in some backstory and society differences along with characters' tendency to make more complex plots and interesting inferences.

Magic is dangerous: There are plenty of ways to kill with first-year spells. The standard advice for those interested in spell creation is to either not have children or wait until those children have grown up. One wizard managed to ruin the magical territories of Sicily. Atlantis somehow got itself erased from Time. This has bred a certain sort of caution into wizarding society, including its fiction.

Magic is restricted: The Interdict of Merlin, intended to help reduce the danger of magic, prevents powerful spells from being transmitted through writing; the key secrets of such a spell can only be passed from one living mind to another. Those who have ancient lore are generally reluctant to teach it. Certain secrets are spoken of only with those who have already worked the secret out for themselves. This has resulted in magical knowledge being lost over time, which is a possible reason behind the blood purists' narrative of magic fading over time.

Magic is different: Considering the above points, of course it would be. Most commonly used spells in canon do something similar in HPMOR, but not all of them are the same.

The Fidelius Charm doesn't exist, or at least is not known to modern magical society.

Time-Turners are strictly limited to 6 turns a day. No person, object, or information can travel back more than 6 hours in a day no matter how many Time-Turners are involved. That said, using a Time-Turner is not dangerous itself even for the full extra 6 hours a day.

Mind magic is generally more limited, although the smarter characters can exploit it anyway. (Obliviation erases memories but not the effects of those experiences. False Memory Charms are a very hard separate spell and require time proportional to the length of the false memory; the proportion decreases with skill. The Confundus Charm is temporary. Legilimency is limited by the caster's understanding of the target, and Occlumency by the practitioner's ability to pretend to be someone or something else.)

Rituals require permanent sacrifices. For example, the Unbreakable Vow requires the vower to sacrifice their capacity for choice on whether to follow the Vow (making it actually unbreakable), the vow receiver to sacrifice their ability to trust the vower on fulfilling the Vow, and the binder to sacrifice some of their magical power.

Transfiguration is not permanent; Transfiguring the wrong thing can lead to disaster when the Transfiguration wears off.

Older artifacts are usually better, but a specialized lesser artifact can beat a generalized greater one.

Economics notes:

The currency exchange rate in magical Britain is roughly 50 pounds to a Galleon.

In the rare cases of marriages arranged with a bride price, ten thousand Galleons is considered a sufficient price for the daughter of a Noble House.