The vulnerability here is tricky to exploit but actually simple to describe. There's a padding oracle in the form of RSA used by both TLS and SSLv2; by repeatedly sending permuted versions of a ciphertext to an SSLv2 server, you can gradually discover the plaintext†. Both SSLv2 and TLS have countermeasures for this attack. But SSLv2's countermeasures are sabotaged by the crappy ciphers it also supports. In both TLS a…
RSA also has the nice property of being deterministic.
ECDSA may not be, depending on how you implement it. EDDSA is.
This matters a great deal in a world where it's important to assume your hardware may have some adversarial properties. It's much easier for your ECDSA device to purposefully leak your private key than it is for RSA (both because there's an explicit covert channel available, and also because of how much smaller elliptic key-pairs are in practice).
Also, as we prepare for a post-quantum crypto world, this might be a bad time for shorter keylengths.
There are lots of great reasons to use curves, but I think describing RSA as obsolete is a little premature.