Tablets are really convenient when doing any graphic design related task, and basically required if you plan on putting any serious effort into redrawing as doing it with a mouse is extremely time-consuming.
As for what tablet you should get, the long and the short of it is that Wacom is expensive, but high quality and
long-lasting, while other brands, such as Huion, are half the price of
Wacom shit for about 3/4ths of the performance.
Also of note is that in my experience, Huion devices take planned obsolescence
to heart. It wasn't even 3 months after my Huion Kamvas Pro 13's warranty ran out
before I noticed some dead pixels at the top of the screen. Turns out that part
of the tablet ran extremely hot. However some thermal glue and a couple
pieces of aluminium made for a decent enough DIY heatsink to fix that problem.
For non-screen tablets, I highly recommend you bite the bullet and get a wacom. It probably won't bankrupt you, and they're really nice. Usually you can even get a used one, because Wacom just lasts. I used an Intuos Pro for over five years daily and it's still good as new.
As for the screen vs non-screen tablet debate, it's very much up to personal
preference
Personally, it also feels easier to draw long, straight lines with screen tablets.
Make of that what you will.
Parallaxing is a problem in all screen tablets, though it's much easier to fix
in software with wacom tablets. Huion's software is dogshit.
Strongly consider getting a Wacom One if you want a screen tablet, even if it's
a little smaller than most others. The price more than makes up for it.