What do I get from fishing perfectly?

There is actually a benefit from perfect catches, but it's not obvious ingame. In addition to extra star tokens in the Stardew Fair, it also multiplies the skill experience received from catching any fish by 1.4.

As of version 1.4, a perfect catch additionally increases the quality of silver and gold star fish by one level.

From my random fiddling around I believe it affects the size of the fish, which loosely correlates to quality. I think your chance to get a gold or silver starred fish increases as you level up in fishing skill. I'm around 4/5 fishing and frequently get starred fish even if I catch them less than perfectly. I think there might be a bit of random wiggle room on all of these.

One thing that getting a perfect definitely does: Fall festival spoilers

In the fishing booth at the Stardew Valley fair you get a bonus to your star tokens based on the number of perfects you get. 1 or 2 perfects gives you 10 per perfect. 3-6 gives about 150 to 200 bonus tokens total. This is the most reliable way to get enough star tokens to buy the good prizes since even winning the grange gives only 1000.