You know, it's really a balancing act, and it'll depend on your individual circumstances.
Among the advantages are the ability to separate out different roles, so have your analytics and front-end development team using Tealium, and your marketing team using GTM. Completely separate access rights, skill needs, etc.
Then there's the fact that you end up with two routes to live, which might suit your needs. If you put all your DoubleClick and AdWords tags into GTM, then the marketing team can publish multiple changes per day up through their test environments and into production without impacting anything that might be going on in TIQ.
And then you've got the product specialisation angle, in which some people would argue that deploying the Google suite of products (Analytics, DC, AdWords) all via GTM is just a better, more integrated way of doing things. So leave TIQ to specialise in turnkey integration with like 1,000 third parties, and let GTM get on with delivering the Google-specific stuff.
But there's a whole bunch of downsides..
Page performance - you're looking at two tag management libraries downloading themselves, that's not great, there's a lot of code that's basically doing the same thing, and extra network requests means socket congestion..
Then you're going to have to maintain two different skillsets within your organisation to handle the two different applications, and your user access management overheads will be potentially doubled.
If things like Privacy Manager are important to your organisation, then you'll need to find a way of making sure that category-level preferences in TIQ are respected by GTM - if GTM is loading both analytics and marketing tags, and those have separate statuses in Privacy Manager, you'll need to find the right way to get that consent working across platforms.
And in general, you've got two platforms to debug, two lots of complexity, potentially two costs.. Unless you've got a good reason for it, it's not usually going to be useful to have two tag management tools on a page at once..
Some time ago, while working in an agency, I worked on migrating a rather large GTM setup to Tealium iQ.
To start with, we moved their GTM container tag into Tealium iQ, so that we also controlled when it was removed from the site, of course after the migration was done.
It did of course come with lower performance of the GTM setup, but I was actually surprised to see how little, since I myself anticipated a lot worse performance than we got, even though it was a rather big setup, we ran from that GTM container.
So, if we're talking about putting GTM in Tealium iQ, or running the two in parallel, I wouldn't be worried much about doing it.
If we then talked about running Ensighten Manage and Tealium iQ together, well, then I would have to get back to you.
Haven't tried that out yet, but I would have to try that sometime in the future ;-)
I totally agree with @UnknownJ, that you should be prepared for somewhat worse performance, either in both, or one of implemented TMS', of course depending on how you were to do the implementation.