Package managers can be divided into two main categories:
Binary package managers: the software is built on some remote machine and you only get the result of the compilation. The most popular (only?) formats being deb (apt) and rpm (yum).
Source package managers: the software code source is retrieved directly and the compilation is done locally. Some source package managers being emerge, pacman, yaourt, slackpkg, pkgng from BSD, Mac Ports, Homebrew, pip (Python), gem (Ruby), etc.
The main advantage of binary packages is that the installation time is greatly reduced if your internet bandwidth is high enough. Reproducibility is also better as to one version of the package will always correspond to one and only one binary.
The disadvantages are the package size (several times bigger than the source code) and the system rigidity: unlike Windows, binaries on Linux usual embeds hard-coded paths and re-locable binaries (binaries that you can move around) are difficult to generate. I.e. binary package manages usually only work in /usr.
To give you an idea of the different between sources and binaries, the Debian archive is currently a bit more than 1Tb but only 72Gb for the sources! One architecture, for instance amd64, being about 95+92=187Gb (2.5 bigger (1)).
Another problem with binary packages is the fixed compilation flags: some optional features may be disabled in the system package, some modern CPU extensions may also be disabled for compatibility reasons...
A debatable point is the tendency for binary package managers to provide older releases. Indeed, mainly source package managers are providing the latest update for every package soon after each release. However, binary packages tends to be extensively tested before reaching the repositories (after all, they have to compile successfully for all the architectures!).
To help you choose, a common pattern is to use a binary package manager for servers and boxes for which one does not want to spend a lot of time in the configuration process. For a development machine used by a "power user" and where you will need bleeding-edge libraries, source package managers tend to be used more frequently.
(1) 95Gb+92Gb is the sum of the amd64 packages and the "all" packages which are the architecture independent files (multimedia resources, fonts, documentation, etc.)