Does the iPhone 5's bundled Lighting connector / power adapter charge the phone faster?
The charger supplies the power that is demanded from it (up to it's rated output), not the other way around.  So it's entirely possible that the same unit will charge different devices at different speeds.  It's entirely possible that a newer battery will charge faster than an identical but older battery, and also possible that the new phone is allowed to pull more charge because it runs a little cooler (charging generates heat, and one of the metrics for deciding how fast to charge is the heat generated as well as the batteries capacity to drink it up).

Basically, it's really hard to measure, without you do a from flat to full charge with the same unit to different devices with batteries of the same age and condition.  Even then, be aware that the phone doesn't stop charing when it says 100%, that's an artificial figure and your phone will continue to charge a little (the equivalent of a little over 5% more logically speaking), then run from battery a little, then charge a little, then run from battery etc - this is to keep your battery in good condition (the same is true for 0%, it will go lower in order to perform a clean shutdown - the battery figures are only an indication, not an accurate gauge) - the actual methods for each device may also vary as to how they handle this.