First of all, nothing provides a better brief on the semantics of these conjunctions than a good dictionary. L&S has quite exhaustive (if not exhausting) articles on et and ac/atque. You can supplement it with others at hand, for example, the Oxford Latin Dictionary, which is not available online.
In the present narrow context, both conjunctions mean pretty much “and,” but there is important semantic difference. et has unmarked conjunctive force, just like English “and,” while ac is more marked, in a sense binding the two ideas more tightly together, but at the same time introducing contrast between them. Note the internal form of at-que ≈ “and-but.” English clearly does not have such a particle; “but” is even more disjunctive than at.
Are they both semantically and syntactically correct?
Are their respective translations accurate?
Yes and yes.
Why do they use different forms of the same conju<n>ction?
They are using different conjunctions. It is entirely incorrect to think of ac and et as “forms” of some abstract meaning “and,” realized in the English word “and.” Latin and English are independent languages on their own, and there is no one-to-one semantic relationship between any two languages.
I do not think it is possible to answer this question literally; the choice of words is in the mind of the speaker, and we cannot peek in there. It is nevertheless instructive to examine somewhat different semantics of these two mottos.
scientia manu et mente
Here “by hand” and “by mind” are simply joined, and English “and” will convey the closest possible meaning: the manu and mente are, in a sense, nearly equal partners in this conjunction.
scientiā ac labore
Here the “and” translation is also possible, but the Latin semantics is richer. The subtlety introduced by “hard” in the translation is felt; “work” is now perceived somewhat more heavyweight in this sentence. ac has a similar emphatic quality toward the second conjunct. The sentence can be roughly translated “by knowledge and also work,” with the force of almost, but not quite, “by not only knowledge but also work.” Think of ac meaning half “but also,” half “and also” in this context.
I understand that the explanation is a little hand-wavy, but languages are different in their semantic repertoire, and particles, often being fossilized shards from earlier stages of a language, carry perhaps even more subtleties in their meaning than significant words, due to their relative longevity.
