It is pointless to compare these things.
"Ajax" is a process that happens in a browser. This is a call to call the local local server page without updating the "main" viewing area, and then performing various actions with this result (data capture, making changes, changing the existing DOM (adding elements)).
Web services is a server-side thing that allows you to call methods in your code, but this call is actually sent to the remote computer. A web service call is usually also made on the server side.
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