Both contract types are cost reimburseable. One has an Award Fee and the other an Incentive Fee. They are very similar on the surface. Both have their own applications, so its not easy to tell which is most commonly used. The answer to that is "it depends".
The award fee is usually based on a subject evaluation of some component of the work. It is also not usually subject to dispute. There is a performance portion of the contract which describes the area of work to be evaluated, the weighting of importance (if any), and the amount of the fee that can be earned each contract period. This contract works best for R&D work where quality is the primary objective. Also for any contract type where the results cannot be objectively measured.
The incentive fee is usually based on objective criteria (often cost-based) that are included in the contract. When cost based, every $ over budget means a greater loss of incentive fee and every $ under budget means a greater portion of the incentive fee is earned. The incentive fee can be based on any criteria such as cost schedule, technical goals (e.g. ship speed), or delivery/schedule.
So in summary, the primary difference is whether or not the contract results are best objectively or subjectively measured. Targeting the desired result keeps the vendor and buyer on the same page. There are many other "rules" regarding these contracts that would be local to the organizations (e.g. the US government has many additional expliciit rules and laws governing the nature of contracts and I have described the common elements to all).