Both wrong and bad have many meanings, and some of their meanings are similar to each other. For the most common meaning, wrong is “erroneous, not correct” whereas bad means, “not achieving a standard, poor”

Wrong usage is incorrect usage. Bad usage is a usage that is not up to standard.

Here are the complete definitions of wrong and bad as adjectives only as per the American Heritage Dictionary:

wrong adjective

1.Not in conformity with fact or truth; incorrect or erroneous.

2.a. Contrary to conscience, morality, or law; immoral or wicked. b. Unfair; unjust.

3.Not required, intended, or wanted: took a wrong turn.

4.Not fitting or suitable; inappropriate or improper: said the wrong thing.

5.Not in accord with established usage, method, or procedure: the wrong way to shuck clams.

6.Not functioning properly; out of order.

7.Unacceptable or undesirable according to social convention.

8.Designating the side, as of a garment, that is less finished and not intended to show: socks worn wrong side out.

bad  adjective worse, worst

1.         Not achieving an adequate standard; poor: a bad recital.

2.         Evil; sinful.

3.            Disobedient or naughty: bad children.

4.            Disagreeable, unpleasant, or disturbing: a bad piece of news.

5.            Unfavorable: bad reviews for the play.

6.         Not fresh; rotten or spoiled: bad meat.

7.            Injurious in effect; detrimental: bad habits.

8.         Not working properly; defective: a bad telephone connection.

9.         Full of or exhibiting faults or errors: bad grammar.

10.       Having no validity; void: passed bad checks.

11.       Severe; intense: a bad cold.

12.       a. Being in poor health or in pain: I feel bad today. b. Being in poor condition; diseased: bad lungs.

13.       Sorry; regretful: She feels bad about how she treated you.

14.       badder, baddest. Slang. Very good; great.

Excerpted from The American Heritage® Dictionary of the English Language, Third Edition  © 1996 by Houghton Mifflin Company.