Prototype

Jquery and prototype are both used for good manipulation of javascript. We can handle difficult tasks of javascript through these. For example working on ajax has become very easy. The json data is also easily handled through them. The jquery uses $ sign while prototype uses $$ sign. We can use noconflict method of jquery and use jquery word instead of $. By default if both jquery and prototype are used in the same project then prototype will dominate on jquery and jquery code will feel problems in execution so I recommend to use jquery instead of $.

Jquery

Prototype is like a framework while jquery is like a library. Jquery has lot of plugins that makes your work easy. Jquery works best for smaller and smart projects. For complex manipulation of javascript prototype is good. But both jquery and prototype will work good. It is easy to shift from jquery and prototype. The syntax of both are neary same. Prototype is near to ruby on rails.  Jquery directly works on dom of javascript. I think jquery is more easy to learn and its syntax is more handy. Prototype is not regulary updated. There is also another javascript library called dojo but it is not as popular as jquery and prototype. Dojo is mostly used with zend framework. But you can also use jquery and prototype in place of dojo. Prototype uses both $ and $$ sign in different cases. If you are new to both these javascript libraries then I suggest you to start with jquery.