Mechanically, I think there are really only three big differences.
First is that spending a point of Willpower in Requiem grants you three extra dice on the roll rather than an automatic success, which is a very big deal.
Next is that botching (or "dramatic failure") just doesn't happen as often in the new World of Darkness system. Dramatic failures only occur when you are reduced to rolling a chance die (one die that only succeeds on a roll of 10) and it comes up a 1.
A relatively minor difference related to that is that success really isn't measured in degrees in the new system. You either succeed or you get an exceptional success. The Storyteller has to declare bonuses or penalties (which grant or remove dice) prior to the roll.
The third big difference is that in combat, damage is figured directly from the result of the attack roll. It's faster, but it can be a lot deadlier.
In terms of the story, the Masquerade setting is gone. Requiem focuses much more closely on local action compared to Masquerade's global tone. Clan infrastructures like the Nosferatu's intelligence network just don't exist, and most vampires align themselves with philosophically like-minded covenants.
If you never played Masquerade as anything but a city game, I don't think you'd find it much different. There are some interesting changes to how the vampiric condition works, though. They get more powerful with age thanks to Blood Potency, so lineage doesn't matter as much. Also, vampires instinctively recognize the Beast in each other using a system called the Predator's Taint.
White Wolf has published a good translation guide for playing Masquerade with Requiem or vice versa.
I'm actually using that guide to run my Masquerade game using Requiem's rules.
Hope that helps!
Best Answer
I would say yes, killing another vampire calls for a degeneration roll and may lead to the loss of Humanity.
Vampires (on the path of Humanity, which we're concerned with here) consider themselves, or at least try to consider themselves (at least partially) human - that is what's reflected by their Humanity score. If a vampire strictly and unquestionably believed that vampires are completely dead as humans, said vampire would have to apply the very same thought to herself too, ie. consider herself dead as a human as well. Were she able to do so, it would mean she would have no Humanity left in her either at all (Humanity: 0.) (Once again, we're dealing with vampires on the Path of Humanity only here.)
As for destroying a vampire with a Humanity/Path rating of 0: I think vampire characters have no means of learning the Humanity score noted on the character sheet of a PC or an NPC. They might very well consider said vampire to be a madman, an utter degenerate worse than the worst of humanity... yet they would consider the vampire in question to be a mad, utterly degenerate... human+Beast combination (see the previous paragraph.) And killing a human, even if that human is a walking horror, may lead to the loss of your own Humanity (as you're getting closer to accepting and utilizing the horror's methods... even if to eliminate the horror.)
If the would-be destroyers somehow gained certain, unquestionable, in-game proof of some other vampire's zero Humanity rating and they gained similarly rock solid proof about the impossibility of vampiric redemption (your ST discarded Golconda?), they could probably kill the zero Humanity vampire without risking losing Humanity for the act. But I highly doubt such proof would be available in the stories of most Storytellers.