What’s the difference between a secret key and a seed? I always hear that if you have the seed then you can easily obtain the key. So, is the secret key just typically an expanded version of the seed? If so, via some key derivation algorithm?

Also, if you can easily obtain the key when you have the seed, why don't we just use the seed as the secret key instead?
I suspect that what you're referring to is the operation of a pseudo-random number generator (PRNG).  It takes a seed as input, and produces a random-seeming series of outputs.  

If you derive a cryptographic key from the outputs of the PRNG, then the key chosen depends on the seed.   If an attacker knows the seed, they may well be able to (much) more efficiently guess the key.

If the seed is created using a source of randomness (as compared with a source of pseudo-randomness), then, at least in part, it is not predictable, and the resulting outputs of the PRNG are not predictable.  If you then throw away the seed (so that it can't be stolen after the fact), you should be able to use the key with some confidence.

You ask specifically:


  So, is the secret key just typically an expanded version of the seed?
  If so, via some key derivation algorithm?


I would say that while it is worded a bit oddly, you could probably look at the secret key as an "expanded version of the seed."  The process is slightly more complicated than just via the key derivation algorithm, because you omitted the PRNG.  So: seed -> PRNG -> key derivation algorithm -> key


  Also, if you can easily obtain the key when you have the seed, why
  don't we just use the seed as the secret key instead?


The short answer is that the seed has a small amount of real randomness that takes time and effort to acquire.  By contrast, the PRNG changes that seed into significantly more pseudo-random output very quickly / cheaply.

You can definitely use the seed (or, a series of seeds, presumably, since you probably need more data than just one seed contains) as a data source to generate your key.  (seed1 + seed2 + seed3 + ... + seedN) -> key derivation function -> key, omitting the PRNG.

The reason it isn't done that way is because:


True random (as distinct from pseudo-random / PRNGs) sources are much much slower and more expensive
Key derivation functions are potentially very sensitive to any predictable information in the input data.  That means that input data (e.g. your seed) that is even partially predictable is dangerous.  So you would need the seed to be generated completely from a slow, expensive, true random source rather than just having a fraction of the seed be truly random.
You need a significant amount of data, so you need more than just one seed.  See above about each one being expensive by comparison to PRNG output.