The Big Brain Bug is an evolved form of the Brain Bug.
Big brain bugs are very similar to normal brain bugs: they only attack with a ranged acid-looking attack that just slows. However, big brain bugs are accompanied by three Small Spiders instead of two, and, unlike brain bugs, these are capable of spawning more small spiders (but will never have more than three at a time). If the big brain bug is killed, its associated small spiders will die soon after.
Big brain bugs may be more difficult to kill than the smaller ones, since they can constantly spawn more small spiders to distract you. As such, a shotgun may be useful to try to kill the big brain bug as quickly as possible, before it can spawn too many small spiders. Consistent damage can also be useful to whittle it down between small spiders; an SMG or melee weapon may be useful for this approach.