There are 12 missions in Hitman: Contracts, you have to complete all of them on the hardest difficulty i.e Professional Mode and the trophy will pop during the closing scene of Mission 12: Hunter and Hunted. The Difficulty trophies stack, thus you will earn Normal Mode Complete and Expert Mode Complete for completing the game on Professional.
There are 4 variations between the difficulty settings:
Saves: When playing on Normal you will get 7 saves per mission, on Expert you will get 2 and on Professional you get none.
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 bullet hits 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.
Playing on Normal and Expert you will start every level with a Silenced Pistol. However no matter what difficulty you are on, you will always start with the: Silverballers, Fibre Wire and a Syringe.