Premiere is strictly speaking a long format video editor; used for trimming clips and sequences and adding transitions. It's also a good tool for for editing sequences to audio. Premiere has decent tools for tweaking audio however, there are more powerful programs that are better suited to audio production (Pro Tools, Logic, etc.) I tend to see video editing and audio production two separate and distinct disciplines and they should be handled accordingly. My two cents of course.

After Effects, as the name implies, is more for post production. Often used "after" a sequence has been edited and using only a small section of that sequence (short format). Though it could be used to edit in a sort of klunky way. The power of After Effects is in the ability to composite layers of clips (diffuse, specular, shadow and reflection passes for example) and/or use plug-ins to achieve special "effects". Another powerful feature is its' ability to key out unwanted backgrounds and composite this over the desired background (think blue/green screen). You can edit to audio by using the peaks of the audio waveform but generally the audio is not massaged in After Effects.

Final Cut Pro is a sort of hybrid of these two methods. You can edit according to professional methods and add special effects using layers and After Effects plug-ins.

Hope this helps.