Each is based on liability for failure to observe a duty imposed by law; one by agreement, the other by a duty to act and perform in a reasonable manner.
A contract is an enforceable agreement entered into between two or more parties. If one party fails to perform under the terms of the contract, then that person incurs contractual liability. For example: Smith contracts with Jones to purchase an automobile for $10,000.00. At the time set for performance (delivery and payment) if Jones fails to deliver the automobile, he has breached the contract and had contractual liability in the amount of the contract, namely $10,000.00. Similarly, if Smith fails to pay for the automobile, he incurs contractual liability for the loss sustained by Jones. This is, of course, an over-simplification; but it illustrates the nature of contractual liability.
A tort is a civil wrong based on the failure of one to perform ones duty to act in a reasonable manner so as not to harm others. Most torts are the result of negligence (failure to exercise due care) but others are intentional. In the scenario described above, Smith completes the purchase of the automobile, so everyone is happy; however when he drives off the lot, Smith calls his wife on his cell phone to describe the deal. His cell phone conversation causes him to disregard a stop sign and he has a collision with another automobile driven by Jones' wife. The car Jones' wife is driving is damaged and Ms. Jones is injured. In this instance, Smith had a duty to keep his car under control, to maintain a proper lookout, to apply his brakes, to obey the stop sign, and (arguably) not to drive and talk on his cell phone simultaneously. He failed in this duty; and his failure to act properly is a tort, a civil wrong.He therefore has tortious liability for the damage to Ms. Jone's car and for her injuries. Again, this is an over-simplification; but it illustrates the difference between the two types of liability.
Oh to add as a novel item, MOST jurisdictions I know of, including Ohio here, have abrogated what is known as a Breach of Promise to marry lawsuit.
You can no longer sue the person who broke off the marriage plans.
Getting an engagement ring back or not or the woman getting to keep it is state specific on the facts though.
Also, most jurisdictions I know of do not classify breach of contract as a tort, therefore, punitive damages are not available in a breach suit.
In a tort action (lawsuit), punitive damages may be available as an additional remedy, expecially if it is a derivative tort, that is, a civil action based on the criminal conduct of another, such as assault.
Some contracts may not be enforceable. Example being, you may lend your spouse $500.00 to be paid back and it is not, a doctrine called Spousal Immunity does not permit a suit to recover such. I don't have a breakdown of states which recognize this though.
Also, any contract against public policy is void and non enforceable, such as your bookie can't sue you to force to you pay an illegal gambling bet.