The references to pipe-weed make it pretty clear that it was, in fact, normal tobacco.
Tolkien smoked tobacco, usually with a pipe.
He doesn't say for sure that pipe-weed is a strain of nicotiana, but says it probably is. He does say with certainty that it is the leaves of the plant that are smoked (whereas most pot smokers smoke the flowers of the cannabis plant, not the leaves).
He never explicitly describes the effects of pipe-weed at length, as far as I can recall, but the effects he implies seem to be limited to mild relaxation, which is indeed the primary effect of smoking tobacco.
In the same paragraph as the "nicotiana" comment, Tolkien calls it "the tobacco of the Southfarthing".
In The Two Towers, he says Merry pulled out a pouch of "tobacco".
In The Hobbit, he uses the word "tobacco" exclusively, never "pipe-weed".
A letter from Tolkien to a pipe smoker who wrote him asking about what kind of tobacco and pipes the hobbits used.
An entire book written about the subject.
In the "Homeward Bound" chapter of The Return of the King Butterbur brings Gandalf and the hobbits "a wad of uncut leaf" of pipe-weed. A tobacco leaf is huge, and must be cut before it is smoked, even if it will be used to wrap a cigar. Marijuana leaves are small, don't come in "wads", and are not usually smoked (pot smokers smoke the "buds", i.e., flowers, not the leaves).
As it happens, I smoke tobacco via cigarettes, cigars, and a pipe. I've also been known to smoke... um... less "legal" things. As cool as it would be to say that Gandalf smokes pot, or even other intoxicants, this is almost certainly not the case. All the effects of pipe-weed as Tolkien describes them are consistent with tobacco use.
Tolkien was many things, most of them awesome, but he wasn't particularly "hip". When he wrote The Hobbit and The Lord of the Rings, he was a solidly middle class professor at Oxford, with several young children. At that time, marijuana was almost unknown outside of the jazz scene. It, and other smokable intoxicants, like opium, were demonized and incredibly taboo. Tolkien would hardly have promoted the use of illegal drugs, especially among children (the primary audience of The Hobbit, and part of the audience of The Lord of the Rings).
All the evidence points to the conclusion that pipe-weed was just plain old tobacco.