The short answer: Tweaks and IVs play very different roles, so it's not really meaningful to compare them. If you view an encryption algorithm as a black box that takes, say, an IV and a plaintext and produces a ciphertext, there aren't any tweaks in sight. It's not until you break open the black box and look inside that you might start seeing tweaks (and tweakable blockciphers).
The longer answer: Tweakable blockciphers, like regular blockciphers, are a tool that cryptographers can use to build other algorithms (i.e., they are used in some mode of operation). They are almost never used directly.
How the underlying tweakable blockcipher gets used depends on the algorithm in question. For example, in OCB encryption, the message is split into blocks and each block is enciphered using a different tweak (the first half of the tweak is set to the IV, the second half is incremented with each block). In contrast, the Skein hash function (one of the SHA3 finalists) allows internal tweak values to be repeated when hashing different messages. This is fine --- tweakable blockciphers are just a tool, and can be used in different ways.
These details, however, can be safely hidden from a programmer who just needs to use OCB to encrypt things (or Skein to hash things). He doesn't even need to know what a "tweak" is! In contrast, he'd better know something about IVs --- in particular, that he shouldn't use the same IV twice with the same key when using OCB.
So to summarize:
- A tweak is one of the input values of a tweakable blockcipher. TBCs are used under the hood in various algorithms. The algorithm will specify exactly where the tweak comes from and how it's used. None of this needs to be visible to a programmer using the algorithm.
- IVs are usually used as inputs to a "higher level" algorithm. Some algorithms may require a random IV, others will just require that each IV is different, but usually the IV is under the control of the programmer.
This distinction between IVs and tweaks can get a bit blurred when it comes to full-disk encryption, because so-called "wideblock" disk encryption algorithms really do use a TBC directly. In this case, the tweak is set to the IV (which is set to the disk sector ID being encrypted).
That being said, at a high level, they play analogous roles in their respective domains: tweaks provide variability to blockciphers (allowing them to process a given n-bit block in different ways) while IVs provide variability to encryption algorithms (allowing them to encrypt a given message differently each time).