If a clause makes sense on its own, then it is an independent clause (an independent clause will always have a subject and a verb). You do have to put a comma before "but" if it precedes an independent clause.
Tom not only forgot his wallet, but he also forgot his phone
In the above example, "he also forgot his phone" makes sense on its own, thus it is an independent clause and must be preceded by a comma.
Now take this example:
Tom forgot not only his wallet but also his phone
In this example, "also his phone" does not make sense on its own, thus it not an independent clause, and no comma is needed.
For your last sentence, you are correct to omit the comma, because "but also kind" doesn't make sense on its own, and is not an independent clause:
Tom is not only smart but also kind
However, you could change the sentence to make it into an independent clause, in which case you would need a comma:
Tom is not only smart, but he is also kind