In the past I have use ASIHTTPRequest but now there is NSURLRequest. Should we be using the NSURLRequest now? Are there any disadvantages
NSURLRequest has been there since iOS2.0 (see the doc), so it is not precisely new. ASIHTTPRequest has been discontinued, the last update was on 15th May 2011, so you shouldn't use it. AFNetworking is a delightful networking library for iOS and Mac OS X as described by them self, and really makes your live easier, in my case it is the first framework I include in my network required projects.
Which one you should use?, I think this depends on your requirements, with NSURLRequest and NSURLConnection you can do everything, in fact AFNetworking us built on top of NSURLConnection. But how I tell you before, AFNetworking has my upvote.
EDIT post iOS7
If you are targeting iOS7 you should take a look to NSURLSession, check the official documentation, and this tutorial.
EDIT for Swift
Take a look to the Alamofire project for an implementation done in Swift, it is also created by mattt (the main author of AFNetworking).