As someone who spent the past few decades deep in the trenches of the digital transformation of photo and video, I'll add a few things to keep in mind...
You can capture very high quality video (like, big-screen Hollywood cinema quality, if you want) with either a phone or a dedicated camera. There are oodles of options.
In the old days (pre-digital), we were all limited by the cost of film and processing/printing. Now, those limits are gone, you can shoot as much as you want. But instead of shooting lots and lots of film, you'll be generating lots of very big digital files. Those have to go somewhere. And these video files get huge.
Shooting with a phone, they'll initially go into the phone's internal storage space (which is generally surprisingly large, but not unlimited). Shooting on a dedicated camera, they'll initially go on some storage media (typically a memory card, again, large but not unlimited). You can shoot happily for a while, but eventually your phone or stack of memory cards will fill up, and you will need to think about managing your digital video files: offloading from your phone or memory cards to someplace else (maybe a cloud storage system, maybe to your home computer, a laptop or tablet or dedicated storage device while you're on the road). No matter where your files go, you'll need to ensure you have those files backed up (ideally in 3 places) - and by the way, the same is true for your photos, if you care about your stuff (because all hard drives fail eventually). This file management can become a non-trivial task, especially while you're traveling.
Then, you need to figure out what (if anything) you want to do with that video footage. It takes some effort and knowledge to turn your raw footage into something resembling a Rick Steves TV show episode starring you, compete with theme song, narration, special effects and credits. This can be a lot more time-consuming than you may expect. And for the best results, you may need a capable computer (you can do some editing on a phone or tablet, but there are reasons professionals use powerful desktop machines with fast processors, lots of memory, big screens, and gigantic storage drives). Mostly you will need time and dedication.
The good news (and there's lots of that) is that with today's consumer gear, you can do things that not long ago were only possible with million-dollar studio setups. Now you can do (and overdo...) lots of the same things, even on your phone or tablet. The bad news is that this definitely takes time, editing videos is among the most "demanding" in terms of requiring good, updated hardware/software, and there's a serious learning curve that may be daunting. It's a very deep rabbit-hole you can disappear down if you want to really pursue it.
In the old days, the limiting factor was paying for all the film and processing. Now you don't have to pay for that, but you "pay" with your time: turning your raw footage of your trip into a nice presentation that tells a story can take a lot of it. You get to decide how far from "quick 'n dirty" to "slick and beautiful" you want it to be.
Before you commit to one piece of gear (a phone or a dedicated camera), give a little thought to what's going to happen after you have shot all those hours of video footage -- what are you going to do with it? Because the answer to that question will drive many other questions, answers and choices about how you will store your footage (while traveling, and once you get home), what software (and hardware) you're going to use to edit your videos, and more. It's worth thinking it through a bit before you jump on one piece of gear and think you're done.
Pressing the "record" button while pointing the lens at something cool is the easy part. It's what comes next that's the real challenge. You'll see. Have fun (and don't stay up all night). 😎