A block cipher is a family of permutations where the key selects a particular permutation from that family. With a tweakable block cipher both key and tweak are used to select a permutation. Tweak and key are related concepts.
The main difference are the security and performance requirements for a tweak:
Changing a key can be expensive, changing a tweak must be cheap.
Being secure when using attacker chosen keys, or related keys, are not primary security properties of a block cipher. Typically they're analyzed assuming a randomly chosen secret key. Related key attacks are rather academic. For example AES is still considered secure despite related key attacks against it.
Related or attacker chosen tweaks must still be secure. The tweak is often a counter, so tweaks are often related.
One application of tweakable block ciphers is disk encryption. You encrypt each block with the same key, but a tweak that corresponds to the block index. Currently we usually don't use a tweakable block cipher for this, but rather XTS mode, which turns a normal block cipher into a tweakable block cipher.