Major difference in Sekiro is the lack of a stamina bar. You can now swing and dodge as much as you'd like, but you can't stunlock enemies like in a Souls game (where once you get the first hit, you can keep swinging until you are out of stamina, depending on the enemy poise stat).
Instead, you have what is called a posture bar, which starts out empty and fills up as you take damage and block attacks without a proper deflection. If you time your block perfectly, you get the aforementioned deflection, which causes the enemy's posture bar to fill up instead of your own, and you can get the opportunity to go in for an attack depending on the situation.
Your posture bar will recover slowly on it's own, and quicker if you are holding a block stance for an extended period of time. Also, some abilities influence posture.
No more soul levels, and this mechanic has been replaced by an XP system which allows "level-ups" to provide you with skill points to unlock new talents and skills. I personally prefer the Souls style of level-ups, but Sekiro has a more simplistic and accessible approach. Perhaps they did this to keep most players at a similar power level so fights can be balanced much better. But that's not to say that there is no progression at all, as there totally is; it's just a much more gradual increase.
Stealth is also a new mechanic to the genre, and it allows you to sneak up and backstab enemies for an instant one-shot death in most cases. Some enemies require two of these "deathblows". Sekiro is much more agile than the heroes of past soulsborne games, as he can freely jump, wall jump, grapple, swim, and fall large distances.
If you enjoy the genre, Sekiro won't disappoint. Amazing, amazing freaking game.