A run-on consists of incorrectly joined independent clauses or too many clauses.

Example: I left and Paul arrives.
Since no comma appears before the conjunction that separates the two independent clauses, this example is a run-on.
 
Example: I left, and Paul arrived, but we saw each other along the way, and we decided to exchange phone numbers, so we could meet for lunch later that week.
This example has too many clauses.
 
To correct run-on sentences, use a comma and a coordination conjunction or separate the independent clauses into sentences.

Example: I left, and Paul arrived.
Example: I left. Paul arrived.
Example: I left, and Paul arrived. However, we saw each other along the way. We decided to exchange phone numbers, so we could meet for lunch later that week.
 
A comma splice is only different from a run-on in that a comma is used without a conjunction when joining two independent clauses.

Example: I left, Paul arrived.

In this case, no conjunction appears after the comma that separates the independent clauses.
 
To correct a comma splice, use a comma and a coordinating conjunction to join independent clauses.