What are the pros and cons of working on contract basis? How does this affect one's career

It will not affect your career. The only challenge will be the uncertainty. As long as you can manage the work you are good.

You will have less authorization compared to your colleagues. Less job stability Some organisations treat you differently (which you can counter from your knowledge) If you want to increase your salary this is one of the good way. It's a good way to experience the work culture of both org.