Prototype is an early version of a product / service / software. Usually it encompasses all the intended functionality and it is the first working model realization of a theoretical idea. Prototypes are generally used for internal functional testing.
Pilot is when a new product / service / software is given to a smaller group of intended audience for use before complete roll out. It is done to understand the following.
1. Implementation issues before complete roll out.
2. How the product is being used and/or misused.
3. Refine the shortcomings (if any).
4. Get initial acceptance.
In software testing, a pilot is usually referred to as a Beta version, in marketing, it is referred to as a soft launch etc.
An application-oriented question on the topic along with responses can be seen below.
Applause for the respondents - Sreejith PC and Nilesh Akre.
Also review the answer provided by Mr Venugopal R, Benchmark Six Sigma's in-house expert.