Update: The current proTeXt actually contains TeXstudio as an editor, and not TeXnicCenter anymore, but that information hasn't even reached all of their own web sites yet.
proTeXt is a bundle that contains, among other things, MiKTeX, which is one of the two major TeX distributions (the other being TeX Live):
First, proTeXt adds a few independent tools to MiKTeX, notably
TeXnicCenter and Ghostscript.
TeXnicCenter is a LaTeX editor that works quite well with MiKTeX, Ghostscript is software for processing PostScript (*.ps) files.
What makes the difference between MiKTeX's 164 MB and proTeXt's 1195 MB is the fact that MiKTeX is a Basic Installer that loads further packages on-the-fly, when necessary. The MiKTeX included in proTeXt has that ability, too, of course; however, it comes with many (all?) packages included, so it's a "full" install.
If you're completely new to LaTeX and you have a decent Internet connection, I recommend going for proTeXt because it gives you all you need for a start and offers a nice installation-flow manual.
If you kinda know where you're going and you have Internet access whenever you need it, the Basic MiKTeX Install will suffice. You can complement it with an editor of your choice, some are listed in the qustion LaTeX Editors/IDEs, you can see which are quite popular among tex.sx users in our TeX Community Polls.
One last thing about MiKTeX vs. TeX Live: I started out using MiKTeX about a year ago and I have found it quite comfortable; but as time goes on, I'm thinking more and more I might switch to TeX Live soon. Some differences are discussed at What are the advantages of TeX Live over MiKTeX?. I personally am just slightly frustrated with MiKTeX because it crashed a number of times and I had to reinstall. Once again, MiKTeX (within proTeXt) might be a good choice for LaTeX beginners on Windows who are used to the Windows feel, but TeX Live really seems to be more popular among people here, looking at the TeX Community Polls again.
I just unzipped the proTeXt 2.2.4 (current version is 3.0) that I still had in My Downloads, and the biggest portion indeed goes to MiKTeX itself. Here's a breakdown of the folders with their respective sizes:
doc 16 MB
gsv 17 MB (GhostScript and GhostView)
Install 10 MB
MiKTeX 906 MB
Sources 1 MB
TeXnicCenter 4 MB
unsupported 9 MB
Indeed, if you dig further in the folder structure, you get to one huge folder: proTeXt→MiKTeX→tm→packages
, which is 903 MB in size and contains over 2000 files.
If you are happy with MiKTeX, then stay with it.
I do not know PCTeX, and hence cannot say anything about potential advantages, but one important difference is: MiKTeX is open source and available for free, while PCTeX is commercial, cf. PCTeX v6 info.
Mike Renfro linked to a thread in PCTeX webforum where you can read that only some years ago PCTeX had no e-TeX support, a TeX enhancement which is the base for most of today’s LaTeXing and included in MiKTeX. I do not know if this has meanwhile changed in PCTeX.
Mike also mentioned the other big TeX distribution TeX Live. For differences to MiKTeX see What are the advantages of TeX Live over MiKTeX? The question title is tendentious, but some answers are about advantages of MiKTeX over TeX Live.
Best Answer
The basic MiKTeX installs a small set of packages. The complete one installs all of the ones that MiKTeX knows about. As MiKTeX can install packages 'on the fly', the main advantages of the complete install are
If you are using your own PC and will be attached to the internet most of the time then you can happily use the basic installation and add packages 'as needed'.
A bit more detail on multi-user systems. MiKTeX will install packages for each user separately, which is fine for a couple of people using a small set of packages, but would be bad for a system used by lots of people with very varying requirements. Hence for network-type installations a 'full' install or at least a bigger set of packages than the 'basic' is usually preferable.