I believe they are not equivalent. Moskito is more broad and only one sub-family of moskitos are Kuloj (Culex). But I am not sure of it.
Vikipedio has an explanation for this. Basically it seems that both kulo and moskito are technically two different species of biting insect, both of which are in a family called kuledoj. Moskito is the correct name for the annoying biting one that everybody hates, but actually in practice people colloquially use the word kulo to refer to anything in the family of kuledoj.
In my personal experience I believe I hear the word kulo to refer these annoying insects much more frequently. It's an especially good word because it opens up the opportunity for jokes about “la fi·kuloj”.
Kulo is the general term for a biting fly, moskito are specifically those bastardly little blood-suckers everyone hates.
Malgranda insekto kies ino suĉas sangon de mamuloj k birdoj, k kies piko kaŭzas malagrablan jukadon
A small insect whose female sucks blood from mammals and birds, and the bite causes uncomfortable itching.
pikantaj dipteraj insektoj, precipe el la fam. de kuledoj
Flying biting insects, especially from the family Culicidae (Mosquito)