The mixer simply gives you more options. It lets you, for example, plug 16 different sources into an 8 track recorder. You can still only record 8 tracks at a time, but at least you don't have to constantly plug/unplug cables. Most recorders also do not have mic preamps in them. Usually they only accept line level sources (either balanced or unbalanced). Of course you can use a stand alone preamp if you have no mixer, or even if you do have a mixer and simply want a better/different sound. A mixer also may have the options for effects sends/returns, headphone output(s), and other various routing features that can minimize the constant repatching of cables.
When mixdown time comes, you need something with faders to adjust levels and combine the tracks together to get a stereo signal. With just a digital recorder, you have no way to do this.
Computer based recording is a different story. The software does the mixing and routing. The mixer becomes an accessory since it still allows you to leave all of your gear plugged in all the time even if your sound card doesn't have enough inputs (same as the stand-alone digital recorder). You can also premix tracks if needed. If you only have 2 sound card inputs and want to record a drum kit, you can run all of the mics through the mixer and combine them into 2 tracks to output to the computer. Obviously, there is the disadvantage of not being able to make changes after the fact.
Personally, I don't use a mixer any more. I do use a few patchbays to avoid crawling around behind the rack when I do have to repatch though. Really, the only thing I miss is having real faders to lay my greay little fingers on while mixing, but with computer based automation being so easy, even that isn't a big deal to me any more.