it's not that it was illegal - that was just a made up plot point for the movie. Coors wasn't sold east of Texas for the simple reason it was a regional product. As more people bought Coors, Coors was able to expand out farther. Here's a newspaper clipping from Lawrence, KS in 1977 talking about how Coors was beginning expansion into Washington state and Missouri:

Lawrence Journal-World, "Coors Plans Beer Sales Expansion", August 2, 1977

It is illegal to transport liquor across State lines for resale unless you have either a Federal Liquor transportation permit or a trip permit if you're doing it just once. Individual State laws vary but usually if the alcohol is for resale then State liquor tax must be paid when sold. Individual use is fine so you wouldn't be charged with a case or two, but a truck load would land you in trouble, but so will picking up strange brides hitchhiking.

The beer in "Smokey and the Bandit" is Coors Banquet. This beer has no preservatives or pasteurization, so it has to be kept refrigerated during transit. This meant it could only be distributed to a few states near Colorado. This limited availability combined with its reputation as something unique among American beers garnered a lot of fans, creating a demand for underground shipping from people loading up their cars on visits to the Rockies to Frederick Amon, who regularly drove truckloads of it to Charlotte, N.C. to sell at a high markup until he caught the attention of the law.

This Time Magazine article (behind a paywall) from 1974 goes into more depth about the beer's popularity and mystique at the time: http://content.time.com/time/magazine/article/0,9171,908509-1,00.html

The American Film Institute mentions the Time article as part of the inspiration behind the film: http://www.afi.com/members/catalog/DetailView.aspx?s=&Movie=55881

Coors has also just started releasing videos on their history, including this (very) brief one in which the current chairman talks about beer smuggling. https://www.youtube.com/watch?v=EGGbHj86teg