The two posts above say it all.

Private busses (or tourists busses) are rife with theft, maintenance problems, and long waits. They are timed conveniently, sometimes include pickup/dropoff, may be priced cheaper, and sometimes more luxurious. But are associated with the most problems.

Thai Busses ( or government busses) are operated by several private companies, leave and arrive at the bus stations, have regulated maintenance and regulated drivers. You can walk up or prebook, and get a ticket with a seat number. On long journeys (Bangkok-Phuket, for instance), simple meals are provided free at the reststops.

Phuket - Samui bus from the government bus station has several departures. Some connect and change in Surat, at least one each day is a direct. I would assume it's the same on the Samui - Phuket route. Both of these routes are reliable.

The taxi fare to Patong from the Bus Depot in Phuket Town should be about 400-450 THB. Or you can take the 30 THB big busses from the traffic circle (less than 1km from the bus station) from Phuket Town to Patong.