Remember that the Nymph does not agree to go with the shepherd and be his love...she basically disagrees with his every point. The biggest one, however, is that the spring time will lead to fall and winter when rivers "rage" and rocks grow cold. The winter will come on them quickly and with vengeance and the shepherd does not give any indication that he will stick around. Everything he offers is temporary and only available in the warmer seasons (with the exception of the buckles for the shoes and the wool dress).
The shepherd does not ever say, "marry me"--he says "come live with me and be my love" but the connotation of his words and the tone indicate that it is lust, not love he is nurturing. This is why the nymph is "raging" like the rivers and her tone is "cold" like the rocks. She does not trust is intentions, and she is a wise nymph to question him.
If we put the word "rage" in the context of the Nymph's tone and attitude about love, it makes perfect sense. "Rage" has a negative connotation; whereas, "rush" does not. The Nympth is extremely skeptical about love and is pessimistic about love enduring, so her diction reflects this.
While "rush" is something that can be done by any animate force (rivers, winds, people, animals, etc.), "rage" as a verb is distinctly human. While animals can experience rage, the severe anger that accompanies it displays humanity.
Therefore, the author is using the word "rage" here in an attempt to personify the river itself. "Rush" would fit, but it would be a weaker word choice. Think of it this way: Businessmen rush, but fighters rage. Therein lies the strength of this author's word choice. Good authors always pick the better word.
Both words are synonyms and somewhat interchangeable because of the type of movement each word implies. The movement both words imply is furious in nature.
Rage can also describe the fervor or enthusiasm of the poet.