Both uses XML for impllimentaion purpose, more precisely Silverlight uses XAML.
The differences are :
Ajax is a technology which majorly deals about client side implimentation, hitting the server/DB from client side without submitting the page to server i.e postback which reduces the round trips to server and fetches the data very fastly.
SilverLight designed for more user friendly Look and feel, concentrates more on playing Audio,video over the web using rapid streaming of files. Microsoft's product for replacement of Adobe Flash.
Easy and faster implimentation of applications like Games using rich dev environment and controls.
To know more about silverlight please go through the link below: