There are a few issues at work here.
Firstly, there was confusion as to the mission of the Red October. From the Wiki:
Officials in the briefing, learning that the bulk of the Soviet Navy
has been deployed to sink Red October, fear Ramius may plan an
unauthorized strike against the United States. Ryan, however,
hypothesizes that Ramius instead plans to defect, and leaves for the
North Atlantic to prove his theory before the U.S. Navy is ordered to
sink Red October.
So to start with, Ryan was tracking and trying to capture the submarine (as opposed to destroying it) as he didn't believe it posed a genuine threat.
Secondly, it has to be remembered that even though the 'caterpillar' drive malfunctioned, it was still a piece of technology that the US didn't have and could learn from. It still could do things that no US submarine could do.
If the US could figure out how it worked and operated, they could attempt to better it and make it more reliable, thus adding to the strength of their naval fleet.
Some demonstration from the movie of Jones' prowess:
Have I got this straight, Jonesy? A $million computer
tells you you're chasing an earthquake, but you don't believe it, and
you come up with this on your own?
Including all the
Sir, I've got all the...
Relax, Jonesy. You sold me.