I've had a bit of work experience in both sectors, although I'm still young, and pretty new to the workforce. Hopefully my limited advice will help.
Advantages of the private sector:
-More opportunities for promotion as the company grows
-These jobs tend to pay higher
-More prestige, and the private sector is supposedly more efficient
Disadvantages of the private sector:
-Less job security
-Very competitive atmosphere, high pressure environment
-Workers' rights are sometimes infringed upon
-Fewer benefits than the public sector
Advantages of the public sector:
-Job security, it's often difficult to get fired
-More relaxed atmosphere
Disadvantages of the public sector:
-Fewer positions available
See the link below for another comparison. Good luck.