A good place to start is to compare the monomeric units, initiators & catalysts used in the polymerization reactions. Isoprene & 1,3-butadiene molecules are very similar. In fact, isoprene is also known as 2-methyl-1,3-butadiene. Not too surprisingly, both isoprene & butadiene can polymerize the same way (from 1,4-addition reactions), and both can even be vulcanized the same way. It's just that with natural rubber, it's not going to involve the Ziegler-Natta catalyst used in making synthetic polymers.
By the way, butyl rubber is not a homopolymer. It's really a copolymer of isobutylene & isoprene.