There is a distinction between the both approaches. When using server side outside HTTP requests, you retrieve the data to the server. If you want that data to be seen to client, you have to send it with the next request the client does to your server side. With server side requests you can do native HTTP requests and cross-domain requests also.
Client side cross-domain requests retrieve the data directly to the client. You can display that to the client in the same instance when the request returns data but if you want the data on the server side (storing the tweets in db) you have to send them from the client, back to the server. Javascript can't do cross-domain XHR requests. In order to do that, you (or the libs that do that) make some workarounds: Using iframes, using including of JS files that already have the info that you need etc.
If you need to consume a web service, I advice on using the backend as the service client and either use timely pull from the client side, or use some of the "comet" techniques.