I used to not care about sound quality, but I couldn't understand why someone would spend $500+ on a pair of headphones if they all sounded the same. So I bought a pair of Sennheiser HD 588 on recommendation from my step dad who works in the music industry.
My mind was blown. The clarity and the bass was unlike anything I had ever heard. I loved my headphones, but with a 9 foot cord and an open ear design, they aren't exactly the best travel companion. I found a pair of ear buds by the same company (CX 200) and they sounded even better than my headphones. The headphones I paid $20 for (they where on sale) out performed my $179 headphones. I took headphones back and just used ear buds for quite some time. I recently purchased another pair of headphones from Sennheiser, but these ones (HD 202) are much cheaper, their sound is very good for the $30 I paid for them.
To answer the question, ear buds when I'm away from home, and headphones when I'm just hanging out in the apartment.
ugh so much typing on a tablet, please excuse any typos