Battery Charging Question!!!!

Well charging with the phone off is going to be faster because NOTHING is using the battery, but it's not a ton faster especially if it's just sitting idle on your desk.  If you are using the phone while it charges, i.e. playing games, emailing, etc, then you will notice a marked slowdown because the phone is using the battery and trying to charge it at the same time.

As for which one is better?  Personal preference, do you want to be able to use the phone while it's charging? THen leave it on.  If you don't need access to the phone or other apps while it charges then by all means turn it off.