The distinction is not between "independently" and "in succession". The real distinction is whether you are doing both, or you are doing just one.
The number of ways to do one of the jobs is $m+n$ (either do the first job, which can be done in $m$ ways; or do the second job, which can be done in $n$ ways; total number of ways, $m+n$).
The number of ways of doing both is $mn$, because you have $m$ ways of doing the first job, and $n$ ways of doing the second job, and you have to do both.
Say you have $5$ pants and $3$ shirts. If you are giving one piece of clothing away, then you have $5+3$ ways of deciding which piece to give away. But if you are deciding what to wear, you need to pick a pair of pants, and a shirt. That gives you $5\times 3$ possible combinations. You can make the choices simultaneously (they don't have to be "in succession"). The "independence" clause is jut that choice of shirt/pants should not restrict your choice of pants/shirt (so you don't have a plaid shirt and a pair of striped pants that cannot be worn together...). What you choose for job one does not affect what you can choose for job two.
So you have to think about what you are doing. I'm not sure if there are "keywords" that you should be looking at.
See also this previous answer