I’m a student of MixLessons, too, and my understanding is that it’s generally better to mix on speakers than on headphones, but there are always exceptions.
For instance, my room is a strange shape with no acoustic treatment on the walls, and that plays a negative part in my mixes due to my hearing uneven reflective sounds as well as the direct sounds from the speakers.
I was told that one way around this is to mix at quieter levels. That works pretty well for me.
Also, regarding headphones, I recently purchased the Waves NX headphone plugin, which has really helped me with mixing on headphones. The problem for me has always been that headphones always make music sound like it’s too close and more inside my head. This plugin spreads out the sound and gives the effect of depth, too, so it sounds like I’m listening on studio speakers.
I’m still experimenting with it at this stage, so I can’t necessarily recommend it, but it is helping me, that’s for sure.
It’s a tricky business, isn’t it, trying to achieve a mix which sounds good when on multiple speakers, let alone when things sound so radically different from one set up to another.