You can get different sounds from both of those methods, but i like to mic my fender bass amp with a sm57, i run that through my pa system for eq and the pre amps, and send that signal lt the recorder. i really like the tone and you get plenty of signal. Many people like to run through a bass pod. Best thing to do is experiment with different setups.
I'd say that the direct sound without a DI . . .well, it SUCKS. (a technical term for "sucks"). I don't have a lot of bass equipment, so, for "direct" to DAW, I often use a plain ol' POD, V-AMP or, mostly, a GNX2. Mostly the GNX because my VF-160CD has an S/PDIF input and the GNX2 has an S/PDIF output (if your DAW has USB you may want to check out POD or RPX400). I also have a small bass practice amp that has a normalized line out, and often-as-not, I use that. I usually record a simultaneous mic track when I use the amp.