We definitely need at least one tag to cover randomized algorithms (that use randomness to do something) and one for random generators (that produce (pseudo)randomness).
For random generators, we have a tag random-number-generator and a more specialized tag pseudo-random-generator, as well as entropy. Most uses of random-number-generator are about PRNG. I'm not sure if this tag is warranted on this site, but technically speaking it is a different topic.
Inasmuch as random/randomness are warranted, I don't see any possible difference in meaning, only a stylistic choice. I'm for making randomness the main tags since we usually use nouns as tags (“this question is about randomness”). I have no strong feeling as to the ~70 questions with these tags should be disambiguated to one of the two sides of the randomness coin.
Similar to other tag combinations, I think we should in addition to Gilles' answer go
tag algorithms can be added to disambiguate.