I too didn't get Cubs at first, but I have since come to understand. Let me help you. It was a journey I was lucky enough to take, I and would encourage you to do as well.
1. Cubs date back to the golden age of aviation. They connect us to an optomistic, can-do world filled with barnstormers, and aviation record breakers. Undoubtedly, these aviators found themselves in one, and you still can today. That's pretty neat if you think about it. Cubs came out arguably during the highwater mark for aviation. They rolled out of the factory and use the same manufacutring methods as Waco's, Stermans, and DC-3s. For most, these other planes are well out of reach.
2. The story of Cubs involves determined American businessmen that endured repeated set backs of every sort as they grasped at the dream of making aviation accessable to the masses. A feat that was arguably closer then than it is now.
3. Cubs will teach you how to fly. Taildraggers with barndoor alerons will make you a better pilot, a skill that you may have noticed is largely missing in modern training if you have happened to have flown commerical lately. Cubs were built to teach you to fly. I recall being told 80% of WWII aviators received their basic training in a Cub. So, it played a key and often forgotten role. THis plane plaed a part in turning farm kids into fighter aces. It is also a warbird itself. It was used by the military for Forward Air Control, an air ambulance, ferrying commanders to the front, and even launched off of barges doubling as carriers.
4. Cubs connect you to some of the greatest aviators ever: Chuck Yeager, Bob Hoover, WASPS, Tuskeege airmen, aeroshell, sean tucker, etc. They all flew them, maybe even the one you happen to get a ride in. I have heard time and again when asked if they could only fly one plane, a Cub is what they would pick.
5. Cubs make friends in a way that carbon fiber and dash full of blinking switches dont. Sit a Cub on the ramp and I gurantee you it will get more requests from folks for a pitcure and more oldtimers walking up to tell you their story than just about anything else. Maybe it is its small size, the yellow paint, the wood prop, the uncluttered dash, or teddy bear like logo, but it is inviting and makes friends. No doubt about it.
6. Cubs were LSA 80 years before LSA. Talk about visionary. With $6 AV gas and skyrocketing costs, a plane that burns 5gph can make new friends even at 60Kts. Sure modern ULs could fill this role and each can, and should, have their fans but to me it is like comparing an Indian to a Honda motorcycle, or a 67 Mustang with a Focus.
Any of the above reasons, seem adequate for icon status. All of them together make it a Cub. It's just your basic warbird, antique, LSA. And more importantly, as it has been since almost the very begining, it represents the hope of GA. Or, as my wife likes to say, it is a flying ray of sunshine.
Thomas, you're definitely right when you point out the Cub's flaws.....but the plane is more than the sum of it's parts or attributes.
If you've never flown a Cub, you need to. Even if you step out (or squeeze out, actually) a the end of the flight and are still unconvinced, at least you will be able to better appreciate why it is so loved today. I'm generally not a fan of "elitist" comments like this, but in this case the saying that "If you have to explain why the Cub is significant to someone, they won't get it" is true.
At the time, it was ubiquitous in flight schools, trained 80% of our military pilots, and was affordable to fly. And yes, it was moderately fast. But today it really is about more than just the history. It's about the nostalgia, and particularly in our modern day of performance figures and critical reviews of new airplanes' fuel burn, climb rate, cruising speed, etc, the Cub is a refreshing reminder of why mankind has desired to fly in the first place: it's enjoyable. Plain and simple.
And this is something that any pilot can (or should) relate to. It's not just the 80-year-old veterans who own, fly, and love the J-3, it's younger pilots as well who just want to slip the surly bonds of earth. May I introduce myself as Exhibit A for the defense? I'm in my (very early) 20s and the Cub is and always has been my favorite GA airplane. As a 10 year old kid, I knew, dreamed of, and loved the Piper J-3 Cub. Simply because it looked like fun. Now, having flown one, I've discovered that I was right.
Go find a Cub and fly it. You won't regret it.