Pick what you know, though from what you have posted I'm not sure you know either of the systems you mentioned very well. Actually, unless you are a professional systems administrator then even picking what you know may not be good enough. You can setup your own servers if you are running on a low budget, but ideally you would be paying someone to take care of all your administration needs.
So, I assume from your budget you won't be buying your own servers and even if you were interested in buying your own servers you don't have the expertise to set them up properly. You will then be looking at hosting.
You should probably be looking at a hosting provider which offers managed hosting. This will be your support. Don't worry so much about the distribution, just choose whatever that provider has available.
Arguably Solaris is superior to Linux in many ways. If you want to go that route I would suggest Joyent as a provider. Linux is wide open for hosting choices. Find one you like and is within your price range and go with them.
You probably need to start out with a VPS as opposed to a dedicated server. Done right, a dedicated server is generally much more expensive than a VPS. Also, a dedicated server requires a lot of knowledge just to know what to look for.
Your widest choice of offerings for a VPS is linux. Pick one which offers a managed plan and go with whatever that plan ships with. If you have a choice, just go with CentOS. Personally, I prefer Ubuntu, but that's not on your list.