Magic is alive and well in Tamriel
Magic is still just as potent in Tamriel as it has always been. However, the way it has been understood and practiced has changed over time.
Schools of magic are arbitrary and only used for aiding in teaching
The in-game book Proposal: Schools of Magic, found in The Elder Scrolls Online, notes that originally the Mages Guild didn't have any schools of magic. However, they found that an academy in Morrowind found teaching to be more effective by dividing it into eight groupings called "schools".
After casting about for a model upon which to reorganize our teaching texts, I came upon the curriculum of the Shad Astula Academy in Morrowind, where they teach magic in eight different disciplines. Though the divisions between the disciplines, or "schools" of magic, are rather arbitrary, magic being an entirely mutable art, this classification of spellcasting into schools of magic has the advantage of providing students with a structure for easy comprehension of the basics of wizardry. The proof of its value is the fact that Shad Astula graduates novice mages with a practical grasp of sorcery in half the time of our own introductory program.
The eight schools were Alchemy, Alteration, Conjuration, Destruction, Illusion, Mysticism, Restoration, and Thaumaturgy. These school have all existed since Daggerfall, with Thaumaturgy being removed in Morrowind and Mysticism being removed in Oblivion.
Indeed, the text even states that although the Mages Guild will initially organize magic into eight disciplines, it may consider consolidating some in the future, notably Thaumaturgy and Mysticism.
The experienced mage will immediately discern the arbitrary nature of these divisions—indeed, we may eventually find it advisable to combine one or more of these schools, assigning the spells within them to other categories. (Thaumaturgy seems a likely candidate for such amalgamation, and possibly Mysticism, as well.) However, these schools seem to be serving Shad Astula well, so I propose we tentatively adopt them as is, and refine as we go.
Levitation was outlawed
Levitation magic, found in Daggerfall and Morrowind as part of the school of Alteration, is absent in Oblivion. This is due to the Levitation Act, forbidding its use. This is referenced in Oblivion as one of the things that citizens say about Dovyn Aren.
"He's getting older, but he can still teach a bit about Alteration. He's been teaching it since before the Levitation Act of [3E] 421."
Note that this act was passed before the events of Morrowind in 3E 427. But because Morrowind is self-governed as part of the conditions of Armistice, they were not subject to this law banning Levitation. However, it was banned in the city of Mournhold (in the Tribunal expansion), so it isn't unprecedented even in Morrowind.
The novel The Infernal City* takes place in 4E 40 and shows that levitation magic, while pretty much forgotten, is not impossible. A Synod mage named Lazarum is said to know of a levitation spell and two individuals used a levitation potion brewed with the ichor of a Winged Twilight.
(Out of universe, levitation was removed because it simplified gameplay. Quests like Caught in the Hunt involve you being trapped on an island or other isolated location, and would be trivially easy to bypass if you were able to levitate away).
* In my opinion, the two Elder Scrolls novels are terrible and should never be read. You have been warned!
Mark and recall appear to just have been removed for gameplay reasons
I cannot find any in-universe reason for why Mark and Recall (and similar teleport-to-location spells like Divine Intervention or Almsivi Intervention) are absent. My guess is that they still exist, just that they are irrelevant for gameplay reasons due to fast travel and the desire to simplify quest design (again, quests like Caught in the Hunt would be trivially easy to break if you could magic your way out of an isolated area).