When do we use this kind of adjective and noun [eg:science essay (science = noun) scientific essay......?

A noun used as an adjective, to modify another noun (or even occasionally to modify an adjective, as in "ice cold"), is called a noun adjunct. It's not a mistake; it's a recognized linguistic feature. So a scientific essay is just that, but a science essay could also be an essay in the field of science, or it could be an essay written for a science class.

I think this phenomenon is a reflection of the German roots of English, since in German there are words consisting of several nouns strung together, each modifying the next. Since all but the last act as modifiers, it's the last one, the only full-fledged noun in the word, that determines the gender.

Several years ago I read a complaint that the phrase "safety deposit box' was ungrammatical, since "safety" is a noun. It should, the writer said, be "safe deposit box." I don;t know what part of speech he thought "deposit " is.

We use nouns as adjectives frequently. Generally, if a noun has an adjectival form, it should be used.

In your example, either way could be correct depending on your meaning. "Scientific essay" means the essay is scientific. But a "science essay" may just mean that you are writing an essay for a science class--the essay may not be "scientific," but "philosophical" in nature.

I think both of your suggested constructs are correct. If any rule exists, it is that nouns preceding a noun can/should be considered an adjective. Examples:

Yahoo! Answers
Web site
biology essay (for biology class)
biological essay (for a biology class or having to do with biology)
coffee mug
tea cup
desk top
movie ticket
silk parachute
horse meat
game board
word game
cork board
wall clock

Oh, but there is a rule! It's quite simple: adjectives show a quality of a thing (the noun), e.g. the blue car, the heavy bag. In the example you gave you have a noun—you have an essay. So, what qualities do you want to apply to it? Is it a blue essay, a heavy essay, an English essay? No, you have a scientific essay.

"Science" is describing the essay, therefore the adjectival form must be used.

-

there's not really a rule on this. i've noticed the mistake myself, but it's commonly used in the language all the time.