Totally different purposes.
There is a mixer built into your MOTU that allows you to route, internally, any inputs to any outputs. That’s super useful for building up routing chains, insert effects, or using it as a format converter. It’s also, for some folks in combination with the app, enough for doing a show or the typical requirements of studio mixing (e.g. headphone / monitor mixing and routing).
If you want to get more hands-on, an analogue external mixer can add things as @bradfromraleigh said already, like (different) preamps (your MOTU has very good ones already, you’d only want others if you want a specific sound), hardware EQ (not necessarily better or worse, but unless it’s a very good mixer, you can get excellent results from software already), and of course physical faders. Will that help your studio or live use? Hard to say - it depends on how much mixing you do and what it’s for. Mixing is not usually required for recording sources - it’s more for live listening of multiple sources at the same time, or for actual mixdowns when you’re trying to take all the tracks from a project and mix them into a two channel final pre-master. Some people do that in the analogue domain, most do it in the DAW. An analogue mixer won’t really offer much if you do it in the DAW.
Really it comes down to what and why you want to mix and in what domain (hands-on analogue, virtual inside the MOTU, or in post, through the DAW), and whether you feel that the specific piece of hardware has a sound you want. Remember, analogue mixers, even the best ones, degrade the sound slightly every pass through them. Once or twice is fine, but several passes (record, playback and mix -> re-record onto another track, repeat) can build up the noise floor and have other accumulative side effects. It’s a perfectly viable workflow, but you need to know why you’re choosing it to really benefit from it.
In short, if you’re just starting out, an analogue mixer will probably not add much to your workflow unless you need to mix up the whole band at once and hear it, live, and maybe catch a 2-track mixdown or send it to a PA or you don’t want to use the MOTU for headphone mixes or something. Otherwise, just multitrack it to a DAW and do any realtime “performance” mixing in the MOTU.
The other devices we are talking about take advantage of the ADAT ports on your MOTU to give you more simultaneous inputs (and outputs in some cases). They aren’t mixers and don’t affect whether or not you need or would make use of one other than to add more channels that you need to decide how to integrate.
