Great question!
First of all, there's a huge point to be addressed here: the power of the temperance movements. Temperance movements, which had been around since before the Civil War (though not as specifically "Prohibition" movements, since they were usually organized through the Church), were constantly growing in power and following. Between 1850-1860, thirteen states "prohibited the traffic in intoxicating liquors" (O'Hara, 597). During the next thirty years another four states followed. Even so, states gradually peeled away from Prohibition during that period, leaving only three of the original seventeen prohibiting it. Part of the problem wasn't just that the laws were repealed, but also that they were ruled unconstitutional in the courts.
But I digress; temperance movements were extraordinarily good at gaining traction throughout history. The Sons of Temperance group founded in 1842 boasted 200,000 paying members, who had taken the pledge to stop drinking entirely. The Women's Christian Temperance Union was organized in Ohio in 1874, and by 1892 had 150,000 members and several auxiliaries besides, more members than women's suffrage movements at the time could claim. It was considered a moral crusade, and attracted what American Society in the antebellum period (before Civil War) had established as the "moral center" of children, the mothers of the world. Mothers, wives, and the like had become the undisputed leaders in the last quarter of the 19th century, as far as Temperance went.
The most powerful of the groups, the Anti-Saloon League, was founded in 1893 and had managed to take the fight's lead by 1900. The organization essentially revitalized the movement, and they focused on getting prohibition across precinct by precinct, so to speak. By 1908, there were only four states where the League had not been established, and it's American Issue magazine had over 16 million subscribers by 1919. The Ohio Dry Federation managed to hit the scene as well, raising over $450,000 over 1917 and 1918, which was plenty of money for the time.
It was in the context of this rising power that the Amendment was passed. So we know the groups were strong now. Why make it an Amendment?
Part of it has to do with the Constitution itself. The Constitution, at Article I, Section 8, gives Congress the power to "...regulate Commerce with foreign Nations, and among the several States, and with the Indian Tribes;". Problem is, does this definitely apply to their power to regulate intra-State commerce? Could they ban the sale of goods within state borders?
To answer that question, consider the question of gun legislation. The court in 1995 ruled on a similar issue, in U.S. vs. Lopez. It essentially argued that:
the commerce power "extends to those intrastate activities which in a substantial way interfere with or obstruct the exercise of the granted power"
While this ruling was handed down in 1995, it was made in light of precedents set as early as 1853. The main thing you could look at, in this point, and which is what I've been getting to, is this: Kidd v. Pearson.
In Kidd v. Pearson, in 1888, the Supreme Court upheld the constitutionality of a state ban on the manufacture of alcoholic beverages within a state. It essentially guaranteed the right of manufacture intra-state, saying it did not conflict with Congress' power to regulate inter-state commerce.
Noting this, however, one has to ask: if Prohibition were passed as a law, would not that law then be struck down as unconstitutional under the Tenth Amendment? As the Tenth Amendment says:
The powers not delegated to the United States by the Constitution, nor prohibited by it to the states, are reserved to the states respectively, or to the people.
So, in noting that the states had the right to handle the manufacturing of the alcoholic beverages, the Supreme Court also could've set a precedent that would hamstring a Prohibition law, since that law could not regulate the manufacture of alcoholic beverages, nor could it do anything unless it had a substantial effect on interstate commerce (as the court held in 1995, and noted it was holding based on precedent from that far back). Now, even if that precedent were wrong, and the court in 1995 was misinterpreting the precedent of the past, that would only serve to show how unclear it was.
Noting this lack of clarity, and noting the issue of manufacture, Prohibition took the cleaner route: that of Amendment. In making it an Amendment, there was not only a greater chance it would stick around (no amendment had been repealed in the past, if memory serves, though parts of the Constitution had been altered by amendments!), it also helped ensure the Constitution didn't conflict with the law itself. It ensured that Congressional power was sufficient to regulate the trade and manufacture, by directly handing the power off to them and taking it out of the hands of the States. Again, as the Tenth Amendment states, it's only the powers not delegated to the United States that are given to the states themselves, so by delegating the power to regulate alcoholic beverages to the United States, it reserved that right from the states themselves. However, when Prohibition was repealed, that meant that states were still free to do as they wish, because the right had been granted again once more to the states.
Hope that helps, let me know if you have questions!
Sources:
HOLLAND WEBB. Temperance Movements And Prohibition International Social Science Review, Vol. 74, No. 1/2 (1999), pp. 61-69
Henkin, David. Becoming America. S.l.: Mcgraw-Hill, 2014. Print.