Both words are common and have various meanings that you can find in any dictionary. Some of those meanings are completely different. The literal meaning of ‘handle’ (to touch with your hands) gives it a sense of being more closely involved or at a lower level. A boss managing something might have his employee handle the details, for example. It’s hard to comment more specifically without knowing what ‘it’ is.

The word manage is more formal than handle, too. It is more likely to be used in books and formal speech than handle. "Manage" can be used with people, but only to talk about business. You "manage" the people who work for you.