Cloud based telephony is sending voice information over an internet service. VoIP is a low budget cost typically rented monthly. VoIP systems are ideal for young or small businesses, but also for big companies which are paying too much for their on premise phone system.
On premise telephony is the basic way of telephony, which means that there is a direct line between you and the customer. On premise is a lot more expensive than VoIP (Cloud based telephony).