I can completely empathize with this problem.
The first step you need to take is to talk with your wife and present a united front. Anything you do or say won't mean anything if your wife will always give him money when he wants it. I know your wife thinks she is just helping him and being a good mother, so you need to approach this sensitively. Here is a good article on helping vs enabling. What your wife is doing is enabling his bad habits, not helping him. This will only cause him (and you) grief down the road as he never learns to be independent. Try to gently explain this to her, and work with her to come up with a solution that works for both of you that isn't as enabling. Perhaps you continue to give him money, but only a set amount each month. Perhaps you agree to pay for school, rent, and living expenses (food), but nothing else. This is a personal choice that you and your wife need to make together.
Once you've decided on something, discuss it with your son. Be firm and indicate that you will not be changing your mind or giving in. If you're only giving him a certain amount each month, tell him he needs to budget it. If you're only going to pay for the bare minimum, let him know he should find a job if he wants a car or new clothes.
Once you've made your expectations clear, be sure to offer to help him figure it out at first. If you just push him out on his own, he's likely to fail. This is where your helping comes in. Offer to teach him how to budget, or help him apply to jobs. Teach him how to succeed so he can do it on his own.
Above all, good luck. Be patient and loving, yet firm and direct. You can do this and your son can become an independent, responsible adult. I've been there, done that, and so I know. It's not easy but it's possible.