The distance from Cairo to Sharm or Dahab is in the range of 500-600Km, so I am not sure if it is better, on the opposite.
As to buses & taxis from Taba, I am not sure what the prices are as I haven't used them in years. I remember something like 6-8EGP for the bus (which used to leave from Taba only once or twice a day, and I suspect it hasn't changed).
The taxis used accumulate 5-7 passengers and sharing the the cost, which was like 50-70EGP for each passenger- but they leave as soon as the cab is full with passengers. If you are alone and there aren't many tourist to share a taxi, you may have to pay the whole price by yourself- I can only guess that a taxi will try to charge you something like 100-150 US$ if you are alone (bus probably about 2-3$).
If you book a liveaboard with any operator from Israel, they will take care of the transportation arrangements. However, there aren't many so you are quite limited in choice. I don't know if you contact a local operator in Sharm if he can- or is willing- to arrange transportation for you from Taba- but it is worth asking.
Another option worth consideration could be a Taba-Dahab or Taba-Sharm liveaboard. Sure, most of the sites along this route can be also accessed from shore, but some of them are quite hard to get from land (only by camel or 4x4) and you will definitely be the only boat in the area- no "cattleboat scenes" whatsoever.
In many aspects, sites along this route like Ras Mamlach, Ras Abu Galum, Gabr el bint, Nabeq etc are as good as any others in the red sea and by far less crowded, and from a boat you can dive also the more popular sites like Dahab's blue hole or Canyon very early in the morning, before the hoards of divers arrive from Dahab.
It is not "the best dive sites scenario" but it is really far far away from crowds... There aren't many operators doing this line, though. You can try inquiring at Taba's Aqua Sport dive club, or in
www.seaqueen.co.il for more information.