The Mormons totally pulled it off- they are way more whacky with their underwear, baptisms of the dead, the Garden of Eden being in Missouri, everyone getting to be a god on their own planet.....and at the same time, they are way more mainstream with accepting college and encouraging personal success of the members, joining the military and running for poliitical office, celebrating Christmas.
JW's can't have tobacco and Mormons can't have caffeine. The caffeine restriction sounds way more strange. They have a much more colorful history with their multiple wives and the Mountain Meadow massacres (http://en.wikipedia.org/wiki/Mountain_Meadows_massacre). But people identify them with the Mormon Tabernacle Choir and more people know the names of their founders, Joseph Smith and Brigham Young, than have even heard of the likes of C.T. Russell and Joseph Rutherford.
Most people just think Mormons are way more devout Christians than Protestants and Catholics and have no idea about the underwear, and the planet Kolob, and they know of the polygamy as a thing of the past. It is only modern entertainment that is finally shedding a broader light on the teachings of Mormonism to tell people that they are whackos.
In people's minds, JW's have remained in the fringe of strangeness with Scientology. People don't know about them either, but many assume they don't believe in Jesus because of Christmas and because of using the name "Jehovah." Heck, people don't even know that Jehovah is basically the same God that Jews and Christians worship. People think JW's are those (slightly) better-dressed Hare Krishna types that sell books instead of flowers and their kids avoid all the holidays and the pledge of allegiance, and they don't do Christmas, and many confuse them with the belief that they don't seek modern medicine and doctors for their ailments.
And yes, as far as we know, JW's have no past of secret killings, no history of having to leave places like Illinois and Missouri and go to the desert. But despite there being at least twice as many Mormons as JW's in the world, people know way more about being woke up on Saturday morning by JW's at the door than by Mormons. Some have figured out that two men with white shirts and "Elder Smith" on their pocket means Mormons, and that sedans and minivans loaded with men and women and bookbags means Jehovah's Witnesses, but most people don't know the difference.