In English we wouldn't interchange those two words; we use "bite" if the insect uses its mouth, and "sting" if the insect has a body part especially designed for stinging (such as the tip of a scorpion's tail or the back end of a wasp). Snakes bite with their fangs; scorpions sting with their tails. Mosquitoes don't have teeth, but they're sucking blood with whatever it is they use for a mouth; wasps sting with their tails.

A bite happens when an animal or insect uses their mouth only. In the case of the mosquito, you have been bitten by its proboscis. A sting can happen from the mouth, or any other part. In the case of the wasp, you have been stung by the wasp's barb.