Yeast can do both, mating introduces genetic variability and allows yeast to evolve. Budding is asexual and creates daughter cells that are identical to the mother cell genetically. Yeast resultant from budding have one copy of the genome (1n) and are haploid. Post mating, yeast are diploid and have two copies of the genome (2n) which can do recombination and leads to genetic diversity. When times get rough, it can split back to haploid cells. To make things more interesting, a-type haploid yeast can switch to alpha-type and vice-versa!
Schizosaccromyces cerevisiae, non-brewing yeast are known as fission yeast, and reproduce by splitting in two, yes, different than brewing yeast which is known as budding yeast.
Yes, I got turned down at a job at a winery for not having enough yeast experience, and the person that got hired does not brew, and has never touched yeast...gotta love networking!