Why is the 808 kick so popular?
Both the TR-808 and TR-909 drum machines have carved their character into many genres, including house, techno, r&b, electro, and hip-hop. Released in 1980 and 1984 respectively, they didn't get much attention at first, but eventually became two of the most well known and used electronic music instruments, along with the TB-303 and the Minimoog.
One of the main reasons of their (eventual) popularity was their price. They were the most affordable drum machines of their time. You'd need to pay around US$5,000 for something like the Linn LM-1, and Roland released the 808 at US$1,200.
The 808 became a common instrument in the arsenal of many musicians. Affordability helped with its popularity, but its sound sealed the deal. It wasn't well received at first because "it was not real enough" (something similar happened with other early synthesizers, like the Theremin), but as people became more and more used to it, its sound spread through all kinds of music including pop and rock.
It was good, it was cheap, and it was everywhere, so its introduction into the mainstream was inevitable. Big names started using it, both live and in the studio. Some notable early adopters include Yellow Magic Orchestra, Marvin Gaye, and Afrika Bambaata.
The kick (or bass drum, as labeled in the 808) was and still is one of the most popular features of the instrument. I don't know if it was the first drum machine to use a fast frequency swipe to build a kick sound, but it definitely was the one that popularized it.
In my opinion, the reason of the popularity of the 808 kick sound (other than the price and big names using it) is that it just sounds great. It is simple, powerful, flexible, and does an awesome job at doing what a bass drum needs to do.
But what differentiates them from the bass kicks of other drum machines?
Design choices like circuitry and synthesis method will inherit character to the sound: timbre, tone, envelope, every single sound dynamic you can think of will be affected, and there's where you'll find the differences.
The "808 kick sound" means more than just "the kick produced by an 808", it now usually refers to any kick built using a fast frequency swipe. So, in that sense, you could say that the main difference is there. Is your kick a fast frequency swipe? No? Not an 808, sorry.
Isn't there something newer and better?
Newer? Sure, all the time. There are countless ways you can produce a kick-like sound. With some time and imagination you can come up with something that probably has never been done before.
Better? Only you can answer that one.