The main issue is lack of authenticated encryption and in consequence padding oracles.
It turned out that fixing them thoroughly is extremely difficult. Again and again new variants of these attacks have been found and existing countermeasures have turned out to be insufficient. While these attacks are of low impact (involves timing and that's hard to pull off remotely), it's certainly not something you want in the most important crypto protocol.
TLS 1.0 has additional issues, notably the BEAST attack.
All of that can be mitigated, but it's ugly and complicated and you really just want authenticated encryption (AES-GCM or Chacha20-Poly1305), which is only available in TLS 1.2.
A good TLS 1.1 server implementation implementing all the fixes should be "fine". While a good TLS 1.0 client implementation should prevent the BEAST from working.
Note that all of these could run without problems if both side support the encrypt-then-MAC extension (https://tools.ietf.org/html/rfc7366).
It's a lot of "if" though, hence why we prefer TLS 1.2 which has no problems (so far). But usually when nothing do, it is still better to have something than nothing.
