Would it be terrible of me to say that I like them both, just in very different ways. I almost don't see them competing with eachother.
Also, to those that said Star Trek because Battlestar Galactica was cheesy... I don't understand. The camp in Star Trek was utterly untouched by BSG. UNTOUCHED!
I have been watching the brutal decimation of all the Star Trek episodes at http://www.sfdebris.com/
and worried it would make me lose love for the series. But I've recently got my girlfriend to watch a full 6 seasons of TNG and we are on to season 2 of DS9. I have to say I think I am enjoying it more now knowing all the flaws. But I'm weird like that. That, and it's been a half decade since I've watched any of the Star Trek TNG episodes. (Did recently watch DS9 over the last 5 years, haven't touched Voyager but that's next on the list during season 3 of DS9.)
I know people like to pretend they are all the same because they are 'science fiction'.
But in truth, I like Star Trek (esp TNG) because it is inconsistent. It's funny. It's serious. It's interesting. It brain candy that will rot your neurons. Everything a growing kid needs. And that's what I grew up on.
Galactica was great because the story arc allowed for a more realistic portrayal of life. I was getting tired of episodic content because real life isn't like that. I like repercussions. I like things not being solved in 44 minutes. I think both have their place. Galactica was also misanthropic: those writers hated the human race. (And especially Cally!) It was a refreshing and different take. As SF debris above said in the BSG mini-series review, "everywhere that Trek zigged Battlestar Galatica zagged."
At that point in my life I guess I was ready for a change. I caught it mostly on reruns. I didn't start watching until after the final 5 were revealed.