The differences aren't so simply summarized like this; they run much deeper than that. Grammarians believe there are principles of clarity and effective communication; they endeavor to discover those principles and teach others the skill. They militate for high standards and "perfection" in the language, or the closest thing to it, according to those principles, which stress logic, order, and academia. Those who are descriptionists deny that such principles exist and think they are figments of someone's imagination or made up and invented.When I first read about the dinstinction of grammmar into descriptive and prescriptive I was a university student in my first year of studies. I was impressed because until then I thought that grammar books were like Holly Bibles and I really liked the idea of describing language, which is a living thing, rather than setting rules and hinder its change.
Most languages in the world — especially secluded tribes or groups in Africa, China, or almost anywhere — are devoid of the grammarian's effect, and so the linguistic approach prevails unopposed: no guidance — just let it be. These languages are living and they no doubt change every generation, but they don't develop along logical, analytical, or academic lines. English has been influenced for a long time, many centuries, by the grammarian's touch (much earlier than the linguist's), which parallels the development of all the other concurrent fields in science, arts, and philosophy — all happening at the same time. It wouldn't make sense to leave out English as a field of study to develop in the light of all these other developments.
In science, we discovered principles in physics regarding gravity, motion, and inertia. Those principles exist and affect us whether or not we recognize them. In the field of music, Bach, in around 1700, discovered — not invented — the principles of harmony that are best pleasing to the ear. Songs in popular music today that incorporate most of these univeral rules prove to be the most popular and well-liked.
Likewise, the grammarians are discovering rules and principles that aid communication. If they make a rule that doesn't aid, then it's a bad or false rule. It doesn't mean rules don't exist or that following principles is a bad idea.
And so one main difference between the two camps is that grammarians believe there are standards that need to be discovered and then used effectively. (Some of those principles work only in the English language, but a good many transcend languages.) Linguists deny such clear-cut standards exist, but that we just haphazardly learn them from our mother or people and culture around us, Therefore, the grammarian isn't discovering stuff, but just inventing, just making stuff up and forcing other people to accept it, they say. And there can't be a more clear or effective manner of saying things — any way is just as good as any other.
The difference in opinion regarding the existence of principles explains also some of the rift and fallout between Chomsky's American brand of linguistics and the European functionalist brand: Chomsky thinks that people have an inbred capacity to learn languages, which are based on universal principles. He also says that words exist to express and serve the universal principles of thought, which exists in all peoples. Functionalists say that no such principles exist, and that the words themselves give rise to "thought." So it's useless to "perfect" words, the language, or communication.
The "rules"of grammar you speak of — I prefer principles — help you more accurately express your thoughts if you avail yourself of these rules — but they only work if you think. I don't know about functionalists, but I do.
