It's probably a good idea to first read through the first game's Gameplay section. This section explains everything that's new in the sequel.
The Health Bar
Instead of having five "penalty marks" indicating his status, Phoenix's health is now indicated by a bar graph. Depending on the risk involved in your accusations, you will lose more health from the bar for each mistake you make. You can tell how much you'll lose by looking at the glowing section of the bar.
Health does not recharge between court sessions - it remains the same through the entire case. In order to recharge it, you must successfully break a Psyche-Lock.
Psyche-Locks
Starting with Case 2, you will be given a Magatama, which lets you see a person's Psyche-Locks. What these locks mean is that a given person is lying or trying to hide something from you. To make an attempt at breaking through these locks, you must present the Magatama to them during an investigation. You will see the locks appear again, and Phoenix will attempt to deduce what is being hidden. Like a cross-examination, you will lose health for presenting the wrong evidence, and it's quite possible to not have enough evidence to break a Psyche-Lock. Make sure you've exhausted all possibilities before attempting to break a lock.
Upon successfully completing a Psyche-Lock sequence, you will regain 50% of your health. If you run out of health, the Psyche-Lock sequence will stop and you will return to the investigation with a minimal amount of health. Needless to say, it'd be bad if you were to go to court like this.
Profiles
While profiles in the first Phoenix Wright game were mostly for informational purposes, you can now present them as evidence during court sessions and investigations. (Hint: Presenting Gumshoe's profile to him during investigations usually creates an amusing response.)