Before moving forward I should probably mention there are conflicting accounts as to the original explicit intention of the Harrison Act, and whether the actual motives were then well known. In any case, the Harrison Act marked the beginning of a major transitional period in Drug Policy, which lasted until 1970 - when congress dropped the whole "taxing" smokescreen and began passing outright federal criminal drug prohibition laws. However, the 1914 legislation being the topic at hand, passage of the Harrison Act played out as follows....
The act took effect in March of 1915. The treasury department had already begun arresting offenders within 2 months of its passage. By some accounts, some people at the time were perplexed as to why the treasury was treating the law as a criminal prohibition, as they had been under the impression that, despite the tax madates, it was simply intended as measure of narcotics record keeping. Considering the various differing historical accounts, we can see that during the time of its passage, some had perceived the act as a revenue generating measure, while others perceived the act as simply a record keeping measure. Obviously, there were also those who believed the act was intended as, and should be treated as, a criminal narcotics control measure (such as the treasury department, and perhaps the framers of the statute).
In the following years, the interpretation of the Act by the courts was inconsistent; A Federal Judge in Pittsburgh held that the prosecution of drug users was illegitimate, and claimed a patient or user was not required to register under the law, thus he could not be said to possess narcotics illegally. A Judge in Memphis claimed it was okay to supply any quantity of narcotics to an "addict" so long as records were properly kept. The Supreme Court virtually struck a major blow to the Act in 1916, when they ruled that the government most likely never intended "to make the probably very large proportion of citizens who have some preparation of opium in their possession criminal." (source)
The Treasury agents backed off temporarily, due to numerous rulings of the courts which were not in their favor.
However, in 1917, Congress passed a wartime act granting the President the ambiguous power of controlling all "necessaries" for national defense. Amidst an era of the anti-saloon league (ASL) and the alcohol prohibition movement, Congress would (ab)use this wartime power, using it to shut off grain and sugar supply to alcohol distilleries - what clearly followed was a defacto prohibition of alcohol during the short 2~ year period of the Great War. In 1919, the 18th Amendment was passed into law, and alcohol became illegal. Important to note is that in the time following ratification of alcohol prohibition, there came to be a general sense of fear that drinkers would simply switch to narcotics like morphine and heroin. So following the successful passage of the 18th amendment, the focus was shifted back to narcotics.
Later that same year, the supreme court made an unusual ruling regarding the Harrison Act - they basically ruled that - the prescribing of narcotics to an addict, and with no intention of curing him, did not constitute legitimate medical practice. Simply put, the court suggested that it was unacceptable for doctors to prescribe narcotics simply to maintain a patient's drug habit.
It gets much worse: A crucial ruling in 1923, regarding a regulation on prescription alcohol, would set a precedent for interpretation of the Harrison Act, as well as all future drug laws; The Court ruled that "the practice of medicine is always subject to the police power of the state." (what the court refers to as "State" in this case is the Federal Government). Following this ruling, most doctors became compliant with the government's interpretations of legitimate practice.
So, in affect, not only could the casual drug user no longer obtain narcotics without a doctors prescription; but they now would be unable to get a prescription unless they fit the government's moralistic criteria of "legitimately needing it".
In the years from 1914-1924, adults went from having complete oversight and command of their own well being and treatment, to a world where the medical establishment controlled their health & well being, and government dictated medical practice.