Are these programs mainly for making sure you get the file in the right way, like that it's complete and what not, or are they also meant to speed up downloads?

Both. And more.

Here's the reasons I like these kinds of programs:

1) Always downloads the file in binary mode, regardless of the server's MIME type setting for that file. This prevents a common kind of corrupted download caused by Netscape Navigator trying to download an executable binary file as an ASCII text file.

2) Can resume an aborted download, even if you lose power half way through the download.

3) Highly tuned download protocols which can sometimes grab the file faster than IE or Netscape could by themselves.

4) Segmented downloading: For popular files coming from impacted servers, can locate mirror copies elsewhere on the internet and download from multiple sites simultaneously. Thus working around the bandwidth limitations of the server, and allowing you to get the files faster (works only if you have a broadband link).

Does anyone know if LeechGet has this last feature? Didn't see it in the screen shots.