Pros (bus): If you buy a monthly unlimited bus pass, you can save a lot of money on what you'd usually spend on gas.
Pros (taxi): You get where you want to, without making tons of stops. Usually you get the taxi to yourself, so you have a lot of space to yourself. You can also call them whenever you need them.
Cons (bus): The buses are almost always crowded. People get in your personal space -- it's not cool. You have to wait 15ish minutes here every time for a new bus to come, too. If you're running late, now you'll be later. There are also time limits on when you can use the bus. If you have to go somewhere in the middle of the night, the city buses don't usually run here.
Cons (taxi): The prices are outrageous. I once had to pay a taxi almost $40 to get across the city. It was not a long drive, nor would it usually cost anything near that money if you were to drive in your own car. I also found that taxi drivers take advantage of you if they know you don't know where you are. Not cool.
That's been all from my experience.
Public transportation helps to save gas as a full bus uses less gas than if everyone on the bus took their own car. It also helps to decrease traffic congestion, as one bus is better than twenty cars on the road. They usually help save people money. In most cases its cheaper top take the bus than pay for gas.
They can be undependable on showing up on schedule. They usually take longer to get somewhere than if you just drove your car. They can be crowded and uncomfortable. They can be smelly. Their hours are limited. In most cases they are not handicap accessible (there is a few exceptions)
Well, primarily it's a service for the public. I'm not sure what the system is like where you are but the public may pay for the public transportation through their taxes. In which case they will want the service to be reliable, fairly priced and efficient. It may also attract tourism to the place, generating more money for the government.
As for the bad things they may provide: a bad service, delays and therefore a potential general waste of the taxpayer's money.
PROS: save money on car maintenace, fuel, parking. Great exercise as you are walking to reach your bus or train station and then to your other venues.... CONS: public transports usually goes down a route you might not want to go if you want to divert elswhere. Having to share a seat with a smelly stranger during summer. Getting wet walking to and from the bus or train station.
Good: cheaper, less traffic, less stress, make friends, faster, can listen to ipod/read a book....
Bad: Can be disruptive , smelly people sit next to you...
Good: get's you where you want to go as long as you know what buses to take.
Bad: Being on time, and some possible anoying riders.