There are 20 missions in Hitman 2: Silent Assassin, complete them all under the hardest difficulty - Professional Mode and the trophy will pop during the closing scene of the 20th mission (Redemption of Gontranno). The difficulty trophies stack, thus you will therefore earn Normal Mode Complete and Expert Mode Complete for completing the game under professional.
There are three variations when upping the game's difficulty:
Saves: When playing on Normal you will get 7 saves per mission, on Expert you will get 2 and on Professional you get none. However in Mission 4: Tubeway Torpedo and Mission 9: Shogun Showdown you do get a bonus save midway through the mission. You can not miss these and you get them regardless of what difficulty you are playing under.
Health: In terms of how much health you get under each difficulty, it is equal. When playing the game under Professional Difficulty, if you get shot at by a guard, you will incur a greater loss of health than you would do if you were playing under Normal Difficulty. It is hard to quantify the difference between the three difficulties because of the variety of weapons. Some are more effective than others but on Normal, you can sustain approx 12 bullets before you die. On Professional it is around 5 bullets before you are in serious trouble but bear in mind some guns are stronger than others. Finally, health does not regenerate.
Maps: On Expert difficulty you will have a limited version of the in-game map, you will be able to see where everyone is but there will be no colour differentiation as they will be represented in a greyish colour. On Professional it will be even more limited as you will only see the targets, exit points and electrical switches.
On Professional Mode If you go round killing everyone in sight, then you really are pushing your luck. Yes on Normal you can play the game in that fashion but when it comes to playing the game on Professional, I strongly advise that you play it in a stealthy manner.