Does American Sign Language (ASL) Have Different Grammar Rules Than the Spoken and Written English Word?

Yes. ASL is very bueatiful. FYI, there are other types of US sign languages that are very different from ASL; signed exact English (SEE) is mainly used to teach reading.—-Anyway, ASL has a grammer more similar to Spanish than English. I say this because you put nouns before adjectives and articles like "a", "the", and "an" don't exist. For example the sentence, "I read a really thick book all day yesterday." would translate to, "Yesterday, all day, read book thick."
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Beyond that, adverbs exist but aren't really spoken. You wouldn't say you read a "really" thick book. You'd just really emphasize the word "thick" with by moving your hands slightly or forcefully as well as adding facial expression, fast or slow speed, big or small movement and maybe even sound with your lips.
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So much of the meaning in this language is about HOW you emphasize words. And although you can translate anything, many hearing expressions don't translate well to SL and vise versa. This includes stuff like bible verses and idioms like "cut off the lights "; meaning "turn the lights off" not "cut the lights off the ceiling". It's a hearing vs deaf culture thing.

This is one way in which learning English with the PC is helpful, as you can replay phrases again and again till you have mastered them. Since eighty p.c of all internet sites are in English, you can read the maximum amount of the language as you would like after you've gained a basic experience of common phrases and vocabulary.