LFE is a specific channel, the .1 in a multichannel system. Bass management redirects low frequencies from full range speakers to the sub(s). As a general rule, bass management will provide better overall response, even when the L/R mains are capable of low frequency playback, because the front speakers must be positioned for imaging, which may not be optimal for bass response. But, this is very much dependent on the room and equipment.
As for LFE itself, it's really a channel designed for movies, where effects like explosions will be heard over everything else. While a multichannel concert mix may have content in the .1 channel, it's likely no different than you would get using bass management with a 5.0 mix.