For me I use my Mackie CR1601 as a direct input to my Focusrite Scarlett 18i6.
Essentially the 18i6 only has two XLR inputs (Combo Inputs that take XLR mic ins, line in and HiZ instrument ins depending on the connection and the settings I choose for those connectors). The remaining six inputs on the back are 1/4" line in (only) connections. So if I want to use those inputs on my interface for anything OTHER than a direct line input signal (which I rarely if ever use) then I can use the mixer.
So if I want to use a microphone/XLR on those back inputs I plug them into the mixer. The mixer has "Direct Out" 1/4" taps for each channel. I connect those to the 1/4" line inputs on the back of my interface and now I can record those mics directly into Sonar.
Only the first 8 channels on the mixer has those direct taps but I only have 8 analog inputs on the interface anyway (but the 18i6 can be expanded via the ADAT/Optical in with a breakout box like the FR Octopre). If I needed more channels from the mixer though I could configure various Aux Out schemes, use the main outs or even use a splitter to tap the headphone output.
Whatever, that's beside the point and I stick with the first 8 channels from the mixer and with the 18i6 I only really need 6 to cover the line ins on the back of my interface (I can just use the XLR ins on the front for the other two mics BUT for consistency of the mic pres I might instead use the mixer channel taps then plug those into the multi ins on the front as line ins like the ones on the back).
Confused yet?
Real world scenario 1...
I want to use Multi Input 1 and 2 on the front of my interface for say guitar on input 1 and bass on input 2. I do this because the multi ins on the back don't have Hiz and I don't have DI boxes to hook those instruments into the mixer (or into the back of the interface).
So input Multi In 1 and 2 are stuffed with HiZ 1/4" instrument connections.
Where are my vocals or mics going to plug into?
They go into the mixer using the XLR inputs and out from the channel direct out taps into the 1/4" line inputs on the back of the interface.
Now I can have both instruments going directly into the interface AND have 6 mics going at the same time.
Real world scenario 2...
I am outputting from some gear outboard gear that outputs too hot/too cold/too crappy to go directly to the interface inputs which have limited or no controls (the only controls are the "trim" controls on the multi ins and the HiZ toggle). I want to turn the signal up or down or do some input EQing (the Mackie has a nice 3 band EQ on each channel as well as a very BROAD input gain/trim knob AND fader). Instead of using the multi ins (which only have the trim and hiZ) I can use the mixer into the line inputs and control volume better as well as EQ things a bit before they hit tape. I could also insert some outboard gear into the analog chain (if I had any outpoard gear and set things up correctly).
Real world scenario 3...
If I've got a bunch of Phantom Powered mics and want to use them to record more than two tracks... well the only Phantom Power inputs are the two multi ins on the interface. So if I want to use 3 PP mics I'd be out of luck. However on the Mackie I have the extra module that adds 8 extra XLR inputs (total 16 with the built in mixer connectors) all of which have Phantom Power. Now I can use more than 2 PP mics (and up to 16). Also due to the extra module on the mixer I can turn on Phantom Power for the first 6 (or maybe 8... I forget) XLR inputs and connect dynamic mics to the rest (unfortunately PP is an all or nothing affair on this mixer aside from the extra module).
Sooo... more Phantom Power options which is great for recording acoustic guit or drums of whatever.
Real world scenario 4...
I simply want to use the different mic pres on the Mackie which are quite nice (so are the ones on the 18i6 but they are different).
Real world scenario 5...
I want to create a sub mix of something to record. Like instead of recording a bunch of different tracks I can just use the mixer to capture the room or a feed or whatever and mix it on the board and record a stereo track OR I just want to record the output from my stereo (I've got a large tape collection so I can transfer to digital and have control over input levels on the way into the DAW using the mixer).
Real world scenario 6...
I'm a lazy, crippled bastard who doesn't want to have to reconnect/reroute cables all the time. With the mixer I can keep a bunch of gear I use a lot connected and only turn on the mixer and turn up the appropriate channels when I need to instead of hurting/annoying myself every time.
Real world scenario 7...
I COULD use the mixer to route out to other gear for playback of stuff like from the DAW or audio/video players to various sources but I never do that.
Real world scenario 8 (but a kind of useless one IMO)...
On my board I can set up 8 channels in and 8 channels out for "real time" mixing. That's where you essentially get all your tracks set up and edited in the DAW (like you would using a tape machine or ADAT) then instead of doing a digital "In The Box" mixdown with automation and whatnot you record playback of the raw tracks then "ride the faders" as the computer records it all. That's old school and VERY cool but not particularly practical considering what we are capable of now.... AND that process can be emulated digitally anyway using outboard MIDI control surfaces. Still cool as franz though.
The real answer... as you can see... is flexibility. Interfaces are just very limited in what they can do to an audio signal and once that signal hits the interface's a/d converters it's over. Having that extra routing and signal conditioning options is VERY useful.
Of course some mixers have really great/classic onboard effects too that many of VST plugins out there try to emulate (like our Prochannel strip and third party stuff people pay a FORTUNE for to emulate). If you've got one of those boards just having those effects/preamps is worth running signals through. However that's where plugins beat out the boards in a way... beacuse those old boards and preamps are HUGE, rare and prohibitively expensive. They also need to be maintained. You can get a pile of good plugs to kind of sort of do the same job on a decent signal... which then requires a good input signal... which may require good input gear... like a mixing board or preamp.
lol... never ending circle.
Outboard mixers can be extraordinarily useful BUT also a huge PITA, cause unnecessary noise and unless absolutely needed for something specific best avoided (IME). It of course also depends on the mixer. They are all different and do different things. Just buying some random mixer doesn't mean you'll be able to do all (or any) of the things I described. They also generally require a lot of manual study to nail down how that specific model works.
Meh... if you are doing okay WITHOUT a mixer I wouldn't get too tweaked on it.
Cheers.
PS: I am a complete and utter amateur poseur and everyones needs are different.