Well, you'd use menus when you want your audience to browse for and select videos using a menu hierarchy, of course. The opposite is a single-movie DVD, which just starts playing your movie when you put it into your disc player.
A picture compilation is a slideshow, basically, that you create in DVD Architect. You can only add pictures to it. No video. But it can have a musical track behind it.
A music/video composition can include video, but you can't have a single music track playing through as you cut between several videos. You're pretty much limited to one video and one audio track.
Compilations are both pretty limited ways to created features for your DVD. You're much better off creating these featurettes in Vegas or Movie Studio and ported them over to DVD Architect as whole videos.
If you're interested, I go into detail on the how-tos (which I take you step by step through) and whys in my book, available on Amazon. You can also pick up a lot of the basics from my free three-part Basic Training tutorial series, available on this page. http://muvipix.com/products.php?searchphrase=dvd+architect&btn.x=0&btn.y=0
People will cite build considerations in answering your question - but actually there is really only two logical authoring considerations for building optical disc titles.
Menu based or not menu based. Menu works best because it's cause of reason is to allow the viewer control of what they want to see and what they want to ignore. The better the menu design - the better the quality of the end user experience. That is the essence of menu design and authoring practice.