RULES:

If I recall correctly, the only core game rules that change are the linked attribute for resisting drain and the types of spirits available for summoning. The magic sourcebooks have always further rules for sub-specialties within the broader categories, usually specific dice pool bonuses/penalties and role playing restrictions.

LORE:

From a lore standpoint, the disciplines are fundamentally different, to the extent that it's hard to even lump them together as "magic".

The hermetic (mage) approach is a SCIENTIFIC understanding of magic. Magic is a detectable force, with formulas describing its usage, and the scientific method can be employed. Being awakened is a physiological phenomenon that allows detection and manipulation of magic in a way similar to how eyeballs are a physiological phenomenon to detect light. Spirits are beings from another plane that sits in parallel to our own, and summoning them is mostly as formulaic as spell-casting. But spirits are a less well-understood phenomenon, and there are differences within the hermetic tradition because the science is less well understood.

The shamanic approach is essentially ENTIRELY a spirit-based approach you might recognize from traditional Native American belief systems or Japanese Shinto. And that's just the defacto shamanic tradition - the magic sourcebooks basically add different shamanic approaches for most traditional religions (e.g. Voodoo has Loas and Zombies).

The basic unifying theme between shamanic traditions is that an awakened individual is chosen, and their power is granted supernaturally (whereas mages believe it is learned, and there is nothing supernatural about it). The defacto shamanic tradition has a totem that guides you. If your totem is Dog, you may have literal communication with Dog in your dreams as he guides you to live his tradition by being a loyal protector, etc. This makes shamans often behave in ways that do not feel "reality-based" to non-shamans, but the way they act differently is ENTIRELY dependent on the totem they serve (although serve is NOT the right word, nor is worship... follow maybe?).

Physical adepts are basically zen meditators tapping a power deep within themselves, using their full potential in a way others don't (this is a simplification of the whole belief system, but its generally accurate). They don't usually percieve any kind of magic outside themselves.

There are MANY other pathways to magic that go beyond this, all with their own belief systems that actually work. Most of those are related to the shamanic tradition and how it fits traditional religious belief structures. You can find them in magic sourcebooks from past editions of the game.