What is the point of isomorphism concept?

Let's say you have a group of big matrices, maybe they are $10000 \times 10000$ matrices with lots of nonzero entries (so you are dealing with a subgroup of $\mathrm{GL}(10000,\mathbb{R})$). And you have to do a bunch of computations with this group. This is a lot of work to multiply these big matrices over and over.

BUT let's suppose you know this group of matrices is isomorphic to some group that is easier to deal with, like for example a cyclic group of order 50. It is really easy to do computations in this cyclic group (you can just do arithmetic modulo $50$). So if you know the isomorphism, you can do your computations in this cyclic group, and translate your results back to your original setting. Then you don't have to multiply the big ugly matrices together.

A friend of mine once got out of a jaywalking ticket because he showed that the pattern of traffic lights at two adjacent intersections were isomorphic to a finite abelian group. From that point he used well-known results to prove that there were multiple elements in the group which allowed for safe crossing as opposed to only the one that was indicated by the "walk" light.

I doubt the judge understood the argument, but it worked. I'd call that practical and concrete!