I'll make the popcorn! It's at least a week since we've had a "mixing on headphones" debate.
Seriously, there are physiological reasons why, even with the best headphones, your mixes sound different when played on speakers...or even other systems. Human ears have a very different frequency response to sound sources clamped right onto them than they do for more distant sounds. Headphones try to compensate for this but it can only ever be a guess since the exact fit is different on everyone.
Add into this the fact that usually headphones are more worried about sounding "good" rather than "accurate". The same applies to lots of speakers too, but the purpose of proper monitors is to show your music warts and all, not make it sound artificially good.
Having said all that, you CAN teach yourself to mix on headphones. You do a mix that sounds good on your cans, then critically listen on a mix of other systems and decide what you need to change. You've already taken the first step by deciding the headphone mix is less vibrant on speakers. Next time, mix it even brighter and try again. You'll eventually learn what things need to sound like on headphones to also be acceptable elsewhere.
A second caveat though...you mention computer speakers. Try to listen on a bunch of other systems before you make any decisions based on what might be a pretty poor set of speakers...