A fully hosted solution requires little to no hardware at your business to operate, and little to no long term maintenance. However, hosted VoIP phones are dependent on a stable internet connection to work correctly. In short, this is an excellent option due to having less up-front cost.
On-premise solutions have the advantage of hardware being installed at your business. For example, on-premise systems can utilize both internet based (SIP) and traditional copper phone lines. As a result, your business gains VoIP level features without being dependent on an internet connection.