I've wrapped my MR2 and I've painted my MR2, so you're going to get a long, detailed answer from me. Just a warning.
tldr answer: get it painted (or paint it yourself - I did)
Each have their pros and cons. For wrap, some pros:
The cost of the wrap is almost your total cost. You need supplies like cutting tools, knifeless tape, various tuck/squegee tools etc, but this is all less than a hundred bucks.
You can also do it piecemeal, as you have the time. Wrap the roof one day, a fender another day, etc.
There are great YouTube channels to help learn how to do it. Paradox (early videos primarily), CK Wraps, lots of others.
Also, an AW11 is probably a pretty easy car to wrap. Small with lots of flat panels and straight lines. An SW20 is NOT easy. The front and especially rear bumpers are a nightmare.
Now, some cons:
There are going to be seams. They may be almost invisible, or they may look terrible - depends how good a job you do. Those seams have a tiny bit of adhesive exposed. That adhesive is going to IMMEDIATELY pick up dust and dirt. You never see this part in the YouTube videos. You can easily wipe them clean with alcohol, but they'll be dirty again in two days.
You will make mistakes. You will waste at least one large piece of vinyl. Normally you need 60 feet of vinyl for a car. You'll probably need more, even for a tiny little AW11.
Cheap wrap doesn't last. One summer parking outside will be enough to fade/discolor any ebay/off-band vinyl. If you're going to wrap, spend the money to get a good product. 3M is a huge brand but be aware a lot of people in the wrap community feel their quality has declined significantly in recent years. I would be more inclined to look at Avery, Hexis, KPMF, etc. Check out Metrorestyling.com and read reviews.
A wrap job will eventually need touch-ups and small fixes. Corners and edges will lift, or air bubbles will appear.
The best wrap is still going to have a little bit of orange peel, while lesser wrap will have a LOT, and there's nothing you can do about it.
Let's say you get tired of the wrap after three years and want a new color. Get ready to spend a week scraping off all the glue residue that will be left behind.
Now for paint, starting with pros:
It'll look better and last longer. No seams. No peeling or lifting corners or edges. Even a cheap Maaco job can be cut and polished to a mirror-flat finish, better than most modern factory paint jobs. But spend a little more for a two-stage job rather than single-stage. Single stage Urethane paint is a lot better than it used to be, but having a clearcoat gives you the chance to cut and buff to perfection.
Cons: upfront cost. A roll of wrap is definitely cheaper.
Quality: Maaco or local shops might do a very good job, or they might not. Does the car need body work? If you have any experience, filling dings and dents yourself and doing a lot of the prep can save you money and lead to a better finished product. You just never know how well the prep work will be done with budget shops. You might have overspray on your radiator or windshield, etc. There's a reason a paint job from a quality body shop is often over ten grand. A quality job takes a LOT of prep work.