“to catch up to” and “to keep up with” are similar but slightly different expressions. *Note, both your sentences are missing important prepositions. They should read: ①Tom was a long way behind the other runners, but he managed to catch up TO them. ②Tanya was running too fast for Paul. He couldn't keep up WITH her. Other than the missing prepositions, you used the expressions in the right contexts. You shouldn’t substitute one expression for the other. “To catch up to” implies a change in position. In #1 Tom was behind but was able to change his position by running faster. “To keep up with” or in your example “to not keep up with” does not imply a change. In #2, Paul stayed behind Tanya the whole time (or most of the time). Or if he could have been able to keep up with her, that would mean he would have stayed running beside/near her (more or less) the whole time. Generally we do use the expression “to keep up with” to imply that one person is faster/more skilled than another person, but the other person is able to match the more skilled person’s abilities.

①Tom was a long way behind the other runners, but he managed to catch up WITH them. or ①Tom was a long way behind the other runners, but he managed to catch them up. This means Tom was behind the others and was running fast enough to reach the same place as the other runners, so the distance between Tom and the others DECREASED to zero ②Tanya was running too fast for Paul. He couldn't keep up WITH her. This means that Paul and Tanya were at the same level in the race, but the distance between them INCREASED so that Paul got further and further behind. So in summary - CATCH up = get nearer and KEEP up = remain together