Israel has three seas: The Mediterranean sea, The Red Sea and the Dead Sea.
Diving the dead sea appears to be suicidal, so we'll talk about the other seas
Med Sea: there are several diving clubs all over the coast from the very south to the northern border of Israel. There are several nice wrecks, including an Italian WW-II submarine (at 33meters), two missile boats, several steam boats, tug boats and more. Med sea also offers archeological sites such as Cesaerea's sunken roman port and also some special places like diving the grottos of Rosh Ha Nikra.
Water temp in the summer can reach up to 30 degrees (Celsius). When the water gets warm also the Jellyfish swarm in and diving gets...more interesting if you know what I mean.
Red Sea: Eilat. Not as good as Sinai, but encountering Dolphins is rather a common experience, with luck- turtles, eagle rays. There are quite a lot of fish species to see, and some parts of the reef are quite good.
There are many dive clubs all over the 14Km of coast, from which only a few Km are open for diving (the rest is closed because of the commercial port, military base etc.)
Water temp in the summer is about 25-26 Degs. Nowadays it is about 23-24.
In Egypt there is definitely *More* diving traffic than Eilat. Much more. In a single day you'll see more divers in Tiran (or Ras Muhammed) than in a whole week in Eilat. In spite of this, the reefs are by far better in Egypt. Perhaps water condition, currents and commercial pollution has less impact in Egypt.
Recommendation? If you can go to Sinai- go for it. It is by far better than Eilat.
If you're "stuck" in Israel, you'll be able to do some excellent dives and most probably see some interesting fish and other critters too. You may even have some surprises.
Which company (club)? There are several, all of them are of good quality. You can try "Manta" (a.k.a Red Sea Sports), "Aqua Sport" and "Meduza" dive centers. All of them have good equipment, excellent guides and facilities. You can get also daily boats to some near sites in Sinai, such as Coral Island.
Good luck.
P.S. If you're coming to Israel, put a message on this board. There are several Israeli participants who will be glad to assist.