Taverna Nikolas in Agni is a true family taverna and Periklis, the owner is our oldest Greek friend on the island. It is a place that we go to because we are friends and like the atmosphere. Toula's is a place to go to if you like an ice bucket! They also do some outstanding food ... at a price. Agni Taverna is (or certainly was) somewhere between the two. We have had some excellent meals there.
Getting back to Boukari, I agree entirely with Tricia's comments. Kalami Fish Taverna is a great place to sit and the food is good. Vasilis is a very good host and always remembers that we have come a long way to eat there. I too would say that Karidis probably has the edge where the fish is concerned but there is not much in it. Both are outstanding.
Without wishing to sound like I am teaching you to suck eggs, I will remind you that fresh wild fish is more expensive than farmed but worth it. Also, Karidis is the only place that I know where you can buy genuine fresh lobster (as opposed to crayfish) There may be other places but I don't know where.