My family of four is looking for the most comfortable and enjoyable travel between these 2 islands by water. Any alternatives to a ferry?

I can't imagine any travel "by water" between the two that would be "most comfortable and enjoyable" other than a ferry, except to charter a boat. That would be the best. I'm not sure what other options you are looking for.

You could book business class tickets on the ferry for a more comfortable journey. The only other alternative I can think of would be to fly back to Athens and catch a connecting flight to Santorini, but then you will be stuck in the departure lounge between flights.

Thank you - Would you recommend a ferry, catamaran or hydrofoil? What are the advanages and disadvantages?

The car ferry doesn't go from Mykonos to Santorini, you have to take one to Paros then change to another that will take you to Santorini. This is a slower way to get there but might be more comfortable if anyone in your family is prone to seasickness. The car ferries are huge and extremely stable.

The Highspeed catamaran is much faster but rocks a bit depending on how smooth the water is.

There is no hydrofoil between Mykonos and Santorini.

Here is the new 2006 schedule for the Highspeed Catamaran.

http://www.ferries.gr/hfd/