Why do some proteins "use" a beta barrel structure instead of alpha helices in transmembrane space?
As you can see on the wikipedia page you linked yourself that beta barrels have far larger diameters (in some cases up to 40 angstroms as stated in this paper) depending on the numbers of beta sheets that make up the barrel, compared to the diameter of an alpha helix (that is fixed) of 6 angstroms. (source http://www.chem.uwec.edu/Modeling/GGProteinStructures/Pages/alpha_helix/helix.html). Alpha helices are good to anchor proteins into the membrane, or forming small holes that is suitable for compounds like ions, while with beta barrels larger compounds such as sugars, or various drugs can be transported. 

There's a good list of proteins with different functions for both types on wikipedia