What's the difference?
A lot of students ask me the difference for this one as I often use both forms, so I thought it would be a good idea to clear this one up. And it is quite a long explanation because the dictionaries do not provide all the information, plus the information I have given here is based on how the words are defined in the UK and by transport authorities*.
A bus is a large motor vehicle that is able to carry many people - which we call passengers - from one fixed place to another fixed place as described, displayed, or advertised in an official schedule/timetable given by the bus company. For the bus, we usually think of these places as within towns and cities, or from villages that are a short journey away from a bigger place, although this is not always the case. (Buses can also be rented/hired, or more formally, used for private hire, but would still have to keep to some kind of schedule as agreed in a formal document between the bus company and the individual renting the bus.) The passenger then pays to use this bus for a journey (known as a bus fare), which can be in the form of a one-way journey, a day return, a week's travel or even a season ticket.
The quality of bus can vary considerably: it can be in the form of a double-decker (a bus that has two levels to allow passengers to sit upstairs), a very nice bus with very comfortable seats (more on this later), and down to a minibus (of varying quality!) that carries up to ten people. In some very remote parts of the United Kingdom, there is even something called a postbus which, in reality, is a Post Office delivery van travelling from one point to another which can also carry the same number of passengers as a normal van or car from one place to another, but under the same rules as a bus.
In the UK at least, as long as it is a motor vehicle with seats operated by a company that has a licence to carry fare-paying passengers from one specific point to another (which makes this different to a taxi, where A to B can be anywhere) then it is a bus.
And in case you didn't know, a fare is the money that passengers pay when they want to travel from one place to another using all kinds of public transport - buses, trains, boats and aeroplanes - including taxis.
A coach is also a bus, but it is more specific. As a general rule, a coach is a large, comfortable bus - which can even be a double-decker - and its job is to also carry seated passengers, but not really for standing. Bus tickets are usually bought in advance of the journey, although some companies do sell tickets on the day and/or even sold by the driver. These buses are usually for journeys where the passenger may have to sit for a longer time than on a normal bus, so there are usually more comfortable seats, along with a luggage shelf above your head to put your things and also a special compartment under the bus - or even the back of the bus - to put bags that are generally too large to go inside. However, it is not unusual for coaches to be used on normal bus services, and in that situation, passengers would be allowed to stand if there are not enough seats on the bus: this, however, depends on the licence that has been agreed between the bus company and the local authorities. Coaches are usually more popular than buses for private hire, regardless of the distance. A minibus can also have very comfortable seating and space to put your luggage, but since it is not large, it's not a coach.
So to put it all into context. If you are taking a passenger-carrying motor vehicle that officially runs from fixed point A to fixed point B for a relatively short distance, it's most likely going to be a bus (even though it could be a coach that is being used to operate a normal bus service). If you're travelling a longer distance, say over a hundred kilometres, you'll need something more comfortable for that journey, so this vehicle, which is bigger than a minibus, is most likely to be a coach.
*In this case, this information is based on what I learnt and understood while working in marketing for a bus company in the 80s and 90s.