(XP Home Edition though so if u have Pro it might be a little different in handling OP1)
Doesn't make any difference. They are the exact same kernel and OS. The only difference is that Pro contains extra stuff you can install like microsoft IIS (a web server developed by MS, really insecure and buggy. Pretty stupid to try and run a full time webserver on windows anyway) and other network based tools / utilities.
the Computer Management console might be an XP Pro thing only, I'm not sure (I've only used XP Pro, not Home at all).
However, both OS'es are still 'workstation-class' OS'es and both have the same kernel. Both do support multiple processors, I'm pretty sure.
Other than those things I just mentioned, the differences are cosmetic only. (eg. version names/numbers, the boot up screen, etc)
So anyway, if you can't run OP1/1.5 properly, just right click the shortcut or EXE > Properties.
Click the Compatibility tab, check "Run this program in compatibility mode for:" and leave the selection on Windows 95.
Also, another thing that may help, because OP1 is a 16bit program, you can go to a shortcut's Properties (NOT the EXE - it doesn't have this option, who knows why, but shortcuts do), click the Shortcut tab (it should end up there when you go to the properties)
Then click the Advanced button, and you'll get a dialog with two checkboxes. The lower one says "Run in separate memory space." (Un)checking this box might improve the game performance (or may let it run if it didn't before).
Save the settings and test the shortcut.
Also, remember that you can set compatibility mode settings using the shortcut properties (it takes effect for the EXE and all shortcuts to it, but the Advanced button setting is only for that particular shortcut)