The most immediate answer is that the cat was sacred to Bastet, a goddess primarily of the Delta region.
Her city was Bubastis. The cat itself has proven to be a pure descendant of ancient Egypt, and earns the right to be the mascot of the Nile.
Recent archeological digs in the Delta region have located a temple, dedicated to Bastet, which may well be the actual breeding ground where the cat was first domesticated, around 3000 BC.
Bastet was the goddess of the ephemeral joys of life, the "wine, women and song," type of fleeting sensual joy, and her celebrations were big parties with lots of flowers.
The cat was her mascot because of its obvious sensuality.
The deeper reasons relate to the domestic relationship of human and cat, since their household cats were not "fed at the table" the way our modern pets are.
Cats had to work for a living, as farm cats do today. The cats lived on vermin - mice, rats, snakes, and bugs. (Yes, bugs. I have a cat who spends her entire day hunting - and eating - bugs in our backyard, and the vet says that it is quite normal.)
A cat therefore, was a household guardian, and would have been revered on those grounds. My cat has even brought down and consumed a four-inch long locust, and if you think about the kinds of problems they would have had with no screens in windows, having a cat around who would chase down that annoying bug in your bedroom would be important.
Snakes were a constant threat in their homes, and a cat could literally be a lifesaver.
The only dogs were large animals, mastiff and greyhound prototypes, and expensive to keep and maintain - a dog like that is not self-supporting as a cat is.
Small dogs as pets were still hundreds of years in the future. The other pets were monkeys, also expensive to keep and maintain, and more demanding.
So the Egyptians, who were a deeply affectionate and sensual people, appreciated the unique nature of the cat.
The Egyptians were remarkably family-oriented, and cats were valued and beloved members of the household.
I think that had more to do with the ban on killing them than any other factor - it would have been similar to killing a child, from their point of view.