TBH I haven't done much listening with any full size HP with the Nano. I've only used IEMs to compare the HP out to using the LOD into the Pico Slim amp. The Nano can of course power IEMs with no problem but the SQ does noticeably improve with the amp, especially with 3-way balanced armature designs such as the Westone 3. The Westone 3 has huge bass and the amp takes total control and allows the bass to bloom where the onboard amp in a DAP can sound a bit thin in comparison. Just imagine the tiny lithium battery in the Nano and how it needs to power everything in there and last for many hours too.
These are my experiences and not all will agree.
There have been many complaints about the Nano battery life. Apple just released an update for the Nano the other day which allows it to fully shut down rather than just go into sleep mode. The battery life should improve with this new tweak. I never tested my Nano for battery life so I can't say much about that. I try to not let the battery life bar go less than 50%.