I think the first advantage is that it can be very cheap in comparison with using a private car, and you don't have to find a parking space, which you always have to do when you're driving, and it can be very inconvenient when you're in a city. Secondly, it can mean that you don't have to drive your car.
You can leave your car home, and you don't have to fill up your car with petrol and this can save a lot of money. But the disadvantage is that there can be a lot of people on public transport, whereas if you're driving your own car, it's just you and the other passengers in the car so it can be a lot more comfortable.
Secondly, you're not able to go everywhere that you want to go. You can only go on designated routes which might not be the place that you want to go. And thirdly, you have to cope with a lot of other people on public transport, so there could be a lot of pushing and shoving, rude people and bad smells, people eating food and all other things that you might don't like.