I spent the first three years of my playing only learning songs off of tabs. I also didn't really take any lessons on music theory, but maybe watched the odd video for bass playing techniques...
After a few years, while I could play 40-odd songs at various difficulties, I still had no idea how chord progressions worked (I didn't even know how to build a chord), and knew none of the chords or progressions in the songs I had learned via tab until I went back to try to figure it out.
I then spent way too much time learning music theory, but not new songs.
Now I'm working on learning songs again, but now mostly by ear. I'm consciously doing this to better my ear's pitch recognition so I can jam better with others, learn songs quicker, and be able to easily duplicate the lines I hear in my mind through my bass. I also find it helps me to understand a song's structure better - the key/time signatures, the chords, progressions, melodies, and how they all fit together.
As for what I use to learn the songs, I'll just try to play it through Audacity and loop parts as needed. I try my best to learn stuff at normal speed, but if a line is particularly busy, I'll use the software to slow it down without altering pitch.