'o' has two (or more) possible pronunciations: o and ɔ (open and closed), while "au" has only one: o
"Bordeaux" is pronounced the same as "Bordo" would be, and "eaux" is the same as "ô" or "au".
The rules for "o" are a bit complicated but it's mostly always open before two consonnants, like in "dehors" and the first occurence in Bordeaux.
It depends a lot on local accent, but "cosse/Causses" have different "o" sounds in the standard pronunciation, while pose/pause have the same closed sound.
I pronounce "Bordeaux" like in "eaux" but I guess I haven't heard it pronounced very often. I grew up in Greece and I speak French from an early age and haven't lived long in France so there's nuances I miss probably.
Bordeaux should be pronounced like eaux so that part is correct :) I just mean that Bordo would be the same, it's all closed o sounds.
In the north and the south east (Toulouse, not Bordeaux) people tend to only use the open o sound though (especially obvious since Toulouse's nickname is la ville rose which should be a closed o) but that's not standard.
Interesting that you're from Greece, I have met quite a few Greeks who speak rather good French and it's always fun to notice how much of French comes from Greek.
Consider e.g. "deors" (outside) or "mouton" (sheep). Now try saying "de-eauxr-s" and "mout-eaux-n" (with a silent s and n). It's not the same sound.