I only care about score if someone with a lower N7 rating outscores me, which has never happened before.
Lol. Then you either have an N7 lower than 120, or you haven't played with nearly enough people 
That said, on topic. Lol the world has really come to a weird place, when a thread such as this comes up and FTL shows up to say he was obsessed with score but he's not anymore, and Cain claims to be the l337est player around 
Anyway, about myself... I'm sorry Loufi, I didn't answer the poll cause I don't feel much represented by any of those options. I'll explain.
I pretty much go with Caerdon. I've been playing this game for a looooong time now, and for most of that time my first and biggest motivation was to improve as a player. Things have changed lately, as I have understaken a certain path in my real life and also because I think I reached some kind of a platoon where further improvement is quite unlikely and would require way too much time to be worth it. So nowadays, I mostly play for fun. But the habit of trying to kill everything in the blink of an eye has stayed, and so has the rage when the game give me bs XD
So, for most of my ME playtime, I strived to become the best player I could. Now, obviously the definition of what a better player is is completely subjective... But mine, personally, is something like "a guy who makes team games as fast as possible". And if you define a better player by that, score actually is a somewhat useful tool to gauge one's performance, although it should be taken with more than a grain of salt. That is, because it obviously has big flaws like not playing objectives yelding more of it than doing them perfectly, combo scores only going to primers, hosting factor not being taken into account, etc. But, it's nothing that a smart individual can't backtrack at the end of a match and decide if the score output is meaningful or if it's not. So yeah, I do care about score somewhat, in my very own personal way. If I'm in a team with friends, I will try to give my best shot at the game, it's just my nature... Which will translate in me trying to kill things as fast as possible, and as a result getting the highest score I can, always with those limitations I just talked about.
But although it can seem the other way I guess, it's only a competition with myself. If I top the score by 100k but I feel like I didn't play good, I will be pissed off at my performance. On the other hand, if I bump into someone so much better than I am that he beats me by 100k in a meaningful way while I judge that I played at my best, well... I'll just be glad I found someone to learn a lot from, really. Sadly, last time that happened was with Feneckus, and I doubt it's gonna happen again. And yeah it's sad cause the learning part is always the most fun! 
So... Yeah, to sum it up - I'm competitive, yeah, but strictly with myself. With others, and about score, only as it's a direct result of my performance.