Would it be grammatically correct to put an omicron/add an article here as well?
Only if you do it in both languages. It's not a phrase; just a word translation. ;)
As far as their pronunciation is concerned, none. "Ο", however, can be part of the diphthongs "-οι-", which makes an "i" sound, and "-ου-", which makes a "u" sound.
The most fundamental orthography rule in the usage of "ω" is that, whenever a verb ends in an "-o" sound (that is, in the 1st person singular), it's always spelled with an "ω" ;)