Compared to AES, ARX ciphers:

1) Are built from constant time operations, which means they are naturally resistant to side channel attacks (timing, cache, power, etc).

2) Are far simpler in their construction. This makes them easier to reason about and analyze.

3) Related to #2, this also makes them really easy to implement, which means less likelihood of some coding mistake.

Beyond that, most recent ARX ciphers also have a few other advantages over AES. For example, Threefish has a built-in tweak field, which makes using it infinitely easier in practice.

EDIT: In case you're hungry for more detailed explanations, I highly recommend reading the papers for Salsa/Chacha and Threefish. They're very well written, easy to understand even if you don't have a lot of experience with cryptography, and they have sections that explain the design decisions in enlightening detail.

ARX constructions are also easier to tune for high software performance, and generally don't require special hardware support, because all CPUs already have fast ARX operations built in.

That's true, but:

* The CPUs that have better-than-ARX (like, fast constant time multiplication) can do better than ARX

* Fast ARX ciphers are still slower than Intel AES hardware.

I like Salsa/ChaCha more than AES, but there's a reason AES is so popular, and it's not incompetence or a conspiracy.