A professional handyman will generally take more pride in his work and as a small business owner should understand basic business facts: good work equals more work, bad work equal unemployment. You will generally have a better result from an experience handyman. That's not to say you can't find a good part timer out there but they are very elusive. In general, professional handymen understand that quality is key to longevity.
How much should you pay for a handyman? That depends on how much you are willing to pay for good service. How much is it worth to you? Some handymen bid the project rather than offer an hourly rate. If you want a handyman to come out, bring his tools, provide good service for minimum wage, you are setting yourself up for a bad thing. Look for a guy that's been around for a while, is professional, guarantees his work and, more often than not, it will be a good experience. Be wary of those who promise top quality work for rock bottom rates. I have seen many handymen use that catch phrase to get work but it is impossible to provide. Top quality translates to more time on the job and that translates to more man hours. If you're willing to pay for it, great.