There are a lot of advantages of owning a professional camera but Im going to talk about just a few important ones. Ok but Im just going to talk about the DSLR cameras, which I think is the majority nowadays.

Personally I don't own one, Why? because I just don't have the resources to have one just yet? Want to donate your old DSLR to me?

Anyways...

1. Flexibility - Having a professional camera gives you flexibility in terms of lens, flash, battery et. al. If you have the budget you can have different camera lenses for different needs. There are a few point-and-shoot cameras that have this kind of flexibility as well, but not as much as a real DSL. Believe me, I know, I have been using other people's DSLRs and it really gives me freedom to do anything I like on a subject.
2. DOF - It is really easy to get the best DOF on a DSLR camera, and if you want to take pictures that have a good composition, you would need a camera that let's you do it without having to go to photoshop or what ever image editor you have and blur the background and other unimportant stuff on your picture.
3. Lighting - Higher ISO settings, better white balance, better aperture priority. Even without using a flash you can take a better picture with a DSLR.
4. Have it your way - I have tried some point and shoot cameras also and when I try to take pictures sometimes it just does not end up the way I wanted it to be. DSLRs have been a great helpin building up the effect I wanted with just a few adjustments on the settings

In some ways these things can be achieved using a regular point and shoot cameras, but if you are a hobbyist, or would want to be in line with photography, I strongly suggest you get a good DSLR camera and just learn how to use it, and you'll realy see the difference. I mean if there is no big advantage at all, why would professional photographers buy an expensive DSLR when they can do it with a point and shoot cameras? So that's how I see it.

You can only enjoy these advantages when you know how to use a DSLR.