Meaning of both words is similar.
I would say that:
аренда is more general. Term is related mainly to immovable property / real assets, and large expencive vehicles. Means to rent officially something significant (farming lands, house or flat*, office, ship, production building or industrial equipment).
We can't say: взять дом на прокат; взять завод на прокат.
прокат legally means the same as arenda. Can be used in cases of borrowing some equipment for reward or without any payment (from a friend). Can be related to a relatively smal devices (tv-set, camera, laptop, smartphone, lamp) or other things like cloths, pan. And all staff what you can ride on - snow/skate/long-board, bike, ski. Прокат stems from прокатиться/покататься.
It's not nice but still possible to say: взять арендовать (взять в аренду) велосипед/машину/миксер.
* for renting a flat(room) we frequently use stable phrase: снять (to hire, to rent from smb) квартиру or сдать (to rent out, to let out) квартиру (комнату)
Аренда is more official and modern language. Прокат is more from historical and specific. In 50-90 years in Russia we had special places (Пункты проката) where people could borrow things for some short period of time (week, month): bycicles, skates, equipment for skiing or swimming, some instruments, a tent, a pump. We still have such services at skating rings, swimming pools or near surfing/skiing resorts - were one can borrow equipment for a day or for a stay. If we use such services we can that: взяли лыжи/коньки/доску/велосипед/костюм на прокат. And as was mentioned above: прокат stems from прокатиться/покататься and usually used with things which you can ride on.