There are two primary consequences of breaking a contract early:
The first is that it can paint you as untrustworthy. The purpose of a contract is to define what services will be provided and what payment there will be for those services. For example, if you go to the bookstore to pick up a brand new book by your favorite author, the store might have sold all its copies. You'd likely come back to the store in the future. If, however, you reserve a copy of the book beforehand (a less formal "contract"), and they do not hold a copy for you, how likely are you to reserve (or even buy) another book there? Would you tell your friends about the problem?
The second issue depends on your contract. A lot of (most) contracts have a clause covering failure to meet the contracted terms. Breaking a contract (or not meeting the terms dictated) can result in forfeiture of payment for services, payment of a fine, or other penalties (this absolutely depends 100% on the contract). Additionally, there may be laws in your city/country regarding this; this will depend on where you are. It is also important to note that depending on the laws where you are, certain parts of the contract may be unenforceable. Again, this completely depends on what the contract says and what the laws are in your city/state/country.
It is generally not advisable to break a contract. Read your contact to be sure of what the immediate consequences will be, and consider whether the risk of a reputation hit is worth abandoning your client for a different opportunity.