They are close synonyms. "Rush" is a common, everyday word. "Haste" is more literary, more formal, a little old-fashioned. While you are learning, understand the word "haste" but use one of the simpler words, "rush" or "hurry."

Notice that "rush" can be both a verb, a noun, or an adjective.

1) "I had to rush to meet the deadline." (Verb, to rush)

2) "There was a rush at work because of the deadline." (Noun, a rush).

3) "If you really need it by Friday, you can have it--but we will consider it a rush order, and it will cost an extra $210." (Adjective, rush, expedited or sped up).

"Haste" is only a noun. "I had to pack in great haste, and I forgot to pack my toothpaste." The adverb is "hastily." "I had to pack hastily." The adjective is "hasty." "My packing was hasty."

It is only a shade of meaning, but "haste" tends to suggest acting too rapidly. For example, there is a rhyming proverb, "Haste makes waste." However, depending on context, "haste" can just mean "with speed, and "rush" can mean "too quickly."

"Don't wait. You must act with haste or you will miss the opportunity." (Haste is being recommended).

"All of those new houses were rush jobs and the foundations are settling and cracking already." ("Rush" here means they were built in a quick, sloppy... hasty way).

If you know J. R. R. Tolkien's The Lord of the Rings, one of my favorite parts is the part about the long-lived Ents, who are always saying "Hoom, hoom, we must not be hasty," and like to take decades to make careful decisions.

haste: speed in doing somthing , especially because you do not have enough time.

rush : to move or to do something  with great speed.