As the title states, I would like to know the biggest changes between the two systems, both mechanic-wise and story-wise. Why should I choose one over the other? (And please, don't use "Vampire: the Requiem is still supported by WW" as a argument.)
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!
A few of the prominent changes:
You might choose Requiem if you wanted a game of various competing political factions and greater intra-group conflict. Requiem is also a much "dirtier" game than Masquerade; the glamour of the '90s (almost '80s) vampire world and its global conspiracies gives way to a grungier, more visceral modernity.