The difference between a commemorative and circulating coin?

The two terms sometimes overlap.

Circulating coins are issued for use in general commerce. Designs are usually not tied to a specific event but instead honor a person such as a king or president, or have images thematic of the issuing country - e.g. the goddess Britannia on the U.K.'s 50p coin, or the ship Old Bluenose on the Canadian dime. The designs are normally issued for a number of years and in many cases become standardized for extended periods, e.g. the Lincoln cent.

Commemorative coins are issued to mark special occasions, events, anniversaries, etc. They're usually struck for limited periods. Some are struck in metals like gold or silver that aren't used for circulating coins, or in denominations that don't circulate such as $10. In the U.S., many commemorative coins are sold only in uncirculated or proof condition in special holders and a premium prices.

However, some commemoratives are also issued for circulation. Examples would include the Isabella quarters struck by the U.S. in 1893 as part of the celebration of the 500th anniversary of Columbus' trip to the New World, and the 50p coins struck by the U.K. in 1994 for the 50th anniversary of D-Day. In fact, the U.S. Washington quarter was supposed to be a 1-year commemorative issued for the 200th anniversary of his birth, but the coin proved so popular that it was continued and became the standard design.

"Series" coins such as the State Quarters and Presidential Dollars are technically circulating commemoratives, but the extent of these programs and the fact that they replace or supersede prior designs blurs the line between commemorative and circulating coins.
A circulating coin is one that you are likely to find in your pocket. A commemorative coin can also be a circulating coin and is minted to commemorate some significant current or historical event in the country of origin. There are also many non-circulating commemorative coins perhaps made of a more precious metal and possibly even Proof or FDC which are usually minted for collectors and investors and are worth significantly more than the face value of the coin.
A circulating coin is for use as money. Commemoratives are made for collectors.