What is the difference between Ajax and MVC?

They are not the same. Client side MVC means that a MVC framework is built entirely on the client side. So the server only delivers data to the application. The binding of the data to the models happens client side

Ajax is not a Framework per say. It was invented by Microsoft in early days to simply make asynchronous calls to a server to get data without refreshing a page. So if you're working on a Greenfield project and would like to have a rich client side App or SPA (Single page Application). You don't need Asp.net MVC. Angular JS will suffice. If you're working on a brownfield project, you might use Ajax to make your app more client side friendly. Also, you might enhance it by a client side framewrok such as angularJs

model-view-controller (MVC) is the name of a methodology or design pattern for successfully and efficiently relating the user interface to underlying data models.

Asynchronous JavaScript and XML 

• Update a web page without reloading the page
• Request data from a server - after the page has loaded
• Receive data from a server - after the page has loaded
• Send data to a server - in the background