That is to say, if the heatsink is short, will the processor's heat be lower compared to that of the tall one?
Actually it's the other way round.
What affects the cooling is the overall surface, the material of the cooler and the airflow around it.
Thus said, a bigger cooler is better, at least to some extent. As long as it is not just a solid block, more cooling fins help spreading the heat to the surrounding air. It has to be a good heat conductor (copper is better than aluminium). And the attached fan, if there is any, should provide good airflow through the cooler. Dust hindering the airflow is bad as well.
I should add that there is no point in purchasing a bigger heatsink if the one you are using is sufficient. Meaning, unless you actually think the specific compontent has a heat-problem or you want to start overclocking, there is no need to switch.
How would you define "sufficient"? Wouldn't cooler always be better, at least as far as component longevity goes? – Ohad Schneider – 2016-11-07T21:46:21.297