An Array{Bool} stores each true/false value as a Bool, which is represented internally as a UInt8. So if your array has N elements, it will take N bytes to store it.
A BitArray stores each true/false value as a single bit, with (conceptually) 8 of them packed into a single UInt8. Consequently, it takes only N/8 bytes to store the array. A BitArray also has methods defined that handle all the required bit-twiddling operations for you.
Depending on the operation, BitArrays are sometimes slower than the corresponding Array{Bool}, and sometimes faster. But by and large the performance differences are quite small, so it makes sense to use BitArrays unless you have a specific reason not to. But overall they are fairly interchangeable.
Note that both are subtypes of AbstractArray{Bool}:
julia> BitArray <: AbstractArray{Bool}
true
julia> Array{Bool} <: AbstractArray{Bool}
true
This makes it easy to write generic methods that take either one.