Unless you're spending upwards of £80 on one, you will not notice the difference with that amp (principally cos it has a dac built in and is a fantastic little unit) and the DT770s which are surprisingly easy to drive. If we were talking some serious high-impedance cans, then you might well need a soundcard.
The 02D is the one with the DAC, the 02 is just a headphone amp.
-snip-
When using headphones from onboard audio though, it's the amplification that can result in lacklustre sound. Proper amplification makes a notable difference. As you've already said, get the headphones and an amp, then go from there.