For being pursuing public accounting you would need certification from CPA or certified public accountant, whereas a private accountant would not need any such certifications. Depending upon the range of clients that you have, you experience would vary when you would be pursuing the job of public accountant, whereas as a private accountant your experience would be confined to the type of industry you are working.
Apart from the basic differences mentioned, in public accounting, the work space would vary from one client to another. Hence, in some places when you can expect to have good people around you, somewhere things would be tough. In case of private accounting, you would be working with your colleagues in a more stable position where tensions would not be that much.
Again, in public accounting, there would be harsh deadlines that you have to follow, and one have to travel a lot, however, there would be no such requirement in case of private accounting. The exposure is much more in public accounting than in private accounting, thus while choosing one; you need to check out your qualities and skills.