Hillerbee is correct your FAA license issued on the basis of the JAA is only valid if you have a current and valid JAA type rating on the type you intend to fly.
It is only the stand alone FAA license that allows you to fly any Rotorcraft category helicopter. Although, you may be legal to fly any of the aircraft in the rotorcraft category, if you had an accident or incident you would need to demonstrate that you had appropriate training to make you competent to fly that type.
So in theory, you could jump straight into a A109 from a robbie legally using a full FAA license, but if you had a prang you would be taken to the cleaners by the solicitors unless you had conducted appropriate training and had appropriate experience.
You also need to comply with the requirements of SFAR No73 to Part 61 if you are going to fly a R22 or R44 under your FAA license.
Where having a full FAA license is handy is for example last summer when I neeeded to position an aircraft. My JAA Typre rating had expired, but I was still able to position the Aircraft under my Full FAA license.
In real terms no-one in their right mind will let you jump into a bigger heli using the FAA license unless you have appropriate experience to that type, even if you have a license which says in principal you can.