Addressing the second half of the question:
The likely goals of a "war on obesity" seem, to me, not to parallel those of the cluster of US policies that "The War On Drugs" usually refers to. So it's possible it would play out differently.
Obesity is a long-term condition, a cumulative effect of things that happen over some period of time. Preventing obesity might involve forbidding or mandating certain moment-to-moment choices (banning chocolate, requiring exercise to use public services) but those things are not themselves the goal. It's theoretically possible to end obesity without eradicating any of the behaviors that promote it.
On the other hand, anti-drug goals are often, if not always, stated in terms of ending drug use (the moment-to-moment choice) rather than drug addiction or other long-term drug conditions. See William Bennett's statement here or a US official boasting about cocaine shortages.
(Note that long-term harms from the use of certain drugs are often cited as a motivation for banning drug use, but that is not the same as using them to characterize what a program's simple intent is.)
For an anti-obesity program to be futile for the reasons the "War On Drugs" is, it would need to be framed as a "war on dessert" and use an all-or-nothing framing under which each time any person eats a donut is a mark against the program, with long-term obesity rates being at best a second-order indicator of success.
I agree that a "war on smoking" would most likely take a form parallel to the War On Drugs, i.e. an attempt to ban and interdict cigarettes, and would likely be framed such that the inevitable market in contraband cigarettes would be treated as the failure of the program, and so would be no more successful than the War On Drugs.
This whole answer is based on speculation about how a "war on obesity" or "war on smoking", so named, would likely be handled by the US government as it contingently exists; I am not claiming any abstract distinctions between the various realms of human behavior that these different "wars" would seek to affect.