Snow Leopard by default will upgrade an installation. If you want a completely fresh installation you need to erase the disk by using Disk Utility from the Snow Leopard DVD (it's in the Utilities menu when started up off the DVD).
Upgrade installations don't most often cause issues but many people recommend against them because of the amount of complexity they add when dealing with issues. Especially when dealing with non-stock software or software that installs in odd locations. In my experience upgrading 7 or 8 Macs it's been a near non-issue with the exception of one computer with a bad kernel extension that required an update before it was stable again.
So no, an upgrade installation is not likely to introduce performance issues on Mac OS X but it complicates how things may have been set up potentially making it harder to find out what the issue is.
Snow Leopard Too Demanding?
Snow Leopard isn't too demanding on the Mac mini you have listed, however the 1GB of memory you have installed is by far your biggest limiting factor when it comes to performance. If you aren't quitting programs then it's quite likely you're constantly swapping memory which is extremely slow. This is exacerbated in Snow Leopard versus Leopard because of possibly loading 32-bit and 64-bit versions of the libraries on Mac OS X. (Regardless of what kernel is used).
So no, Snow Leopard is not too demanding but you're definitely holding it back with only 1GB of RAM.
Fresh Install Better?
Normally yes - it means you have no possible cruft from a previous installation. There are many instances however where it would not make a difference - for example, a badly behaving application once installed on a fresh system will be just as bad as an upgrade.