A premise is a beginning point for any argument, it is used (with other premises and conlusions from them) to demonstrate or deduce certain truths. In good discourse, the premises ought to be acceptable by both parties - in such cases the argument can be judged as valid or invalid without worrying about the specific soundness; as a valid argument with true premises is sound. Take the argument...

All men are mortal. Socrates is a man. Therefore Socrates is mortal.

The first two sentences are premises, they are used to prove the conclusion. Since we likely accept the first two sentences are true, the argument then functions well in discourse.

Now, both theses premises are also assertions - in the context I have not shown them to be true, but assumed them true. As such, you can see there is overlap. In general, however, when someone claims that X is an assertion, they mean to say that it cannot be accepted as a premise and must be shown true by some means.

So, while there is no sharp divide we can find useful heuristics. A premise ought to be accepted for an argument to succeed. An assertion is what ought to be shown - it should be the conclusion of another argument before being used in the present one. So, while a premise is usually an assertion (not shown to be true), there is a linguistic difference when someone points out a premise versus pointing out an assertion - the latter is usually indicative of a bad thing, it has a negative connotation.