The most obvious difference is that aptitude provides a terminal menu interface (much like Synaptic in a terminal), whereas apt-get does not.

Considering only the command-line interfaces of each, they are quite similar, and for the most part, it really doesn't matter which you use. Recent versions of both will track which packages were manually installed, and which were installed as dependencies (and therefore eligible for automatic removal). In fact, I believe that even more recently, the two tools were updated to actually share a databased of manually vs automatically installed packages, so cases where you install something with apt-get and then aptitude wants to uninstall it are mostly a thing of the past. 

There are a few minor differences:

 - aptitude will automatically remove eligible packages, whereas apt-get requires a separate command to do so
 - The commands for *upgrade* vs. *dist-upgrade* have been renamed in aptitude to the probably more accurate names *safe-upgrade* and *full-upgrade*, respectively.
 - aptitude actually performs the functions of not just apt-get, but also some of its companion tools, such as apt-cache and apt-mark.
 - aptitude has a slightly different query syntax for searching (compared to apt-cache)
 - aptitude has the *why* and *why-not* commands to tell you which *manually installed* packages are preventing an action that you might want to take.
 - If the actions (installing, removing, updating packages) that you want to take cause conflicts, aptitude can suggest several potential resolutions. apt-get will just say "I'm sorry Dave, I can't allow you to do that."

There are other small differences, but those are the most important ones that I can think of.

In short, aptitude more properly belongs in the category with Synaptic and other higher-level pacakge manager frontends. It just happens to also have a command-line interface that resembles apt-get.

## Bonus Round: What is wajig?

Remember how I mentioned those "companion" tools like apt-cache and apt-mark? Well, there's a bunch of them, and if you use them a lot, you might not remember which ones provide which commands. wajig is one solution to that problem. It is essentially a dispatcher, a wrapper around all of those tools. It also applies sudo when necessary. When you say `wajig install foo`, wajig says "Ok, `install` is provided by `apt-get` and requires admin privileges," and it runs `sudo apt-get install foo`. When you say `wajig search foo`, wajig says "Ok, `search` is provided by `apt-cache` and does not require admin provileges," and it runs `apt-cache search foo`. If you use wajig instead of apt-get, apt-mark, apt-cache and others, then you'll never have this problem:

 $ apt-get search foo
 E: Invalid operation search

If you want to know what wajig is doing behind the scenes, which tools it is using to implement a particular command, it has `--simulate` and `--teaching` modes that show you.

Two wajig commands that I use often are `wajig list-files foo` and `wajig whichpkg /usr/bin/foo`.