Final answer:
A run-on sentence incorrectly joins two independent clauses and can be corrected by a period or a semicolon. A comma splice is a similar error fixed with similar punctuation corrections.
Step-by-step explanation:
The grammar mistake that mistakenly joins two independent clauses, which can often be rectified by using a semicolon or period, is known as a run-on sentence or a comma splice. A run-on sentence involves two or more complete sentences that are not properly punctuated. To correct it, one could add a coordinating conjunction such as 'and' or 'but', separate the clauses with a period to create individual sentences, or use a semicolon when the clauses are closely related.
Remember that a comma splice also joins two independent clauses incorrectly with just a comma. This too can be resolved with similar methods: adding a period, semicolon, or a coordinating conjunction to separate the thoughts into proper sentences or combining them into a correctly punctuated compound sentence.