Usually a bus is a form of public transport. It's part of a network and owned by the local council or administration, or contracted out to a private company but with the local council still having overall responsibility for the service. A bus has a specific route which it will follow, perhaps several times a day, and generally will pick up anyone who wants to travel and can pay for a ticket. You can buy season or discount ticket for bus travel, but commonly people will just go to the bus stiop, wait for tne next bus to arrive and get on it, paying when they are on the bus.

A coach, on the other hand, is a private vehicle. It's usually going to a specific destination, eg in another city, and you usually have to book a seat on it in advance. This means you have to travel by a specific coach; if you miss it, for instance, you can't usually just take the next one (though it may be possible if there are spare seats.)