I will note, for the moment, that said inconsistency was handled pretty much exactly as it is in the comic books.
DC's Alternate Company Equivalent, The Atom, possesses full control over both his size AND his density -- and, of course, this was spelled out in dialogue, editorial footnotes, or both, in every single one of Dr. Palmer's Silver Age adventures. However, Dr. Pym, Ms. Van Dyne, and their assorted successors have never explicitly been granted anything but size control -- and usually, they are portrayed as no stronger than their insect associates at those scales.
A possible explanation of the Tiny/Heavy Paradox that plagues the movie occurred to me this morning. It's nonsense sci-fi technobabble, but no more than any other instance of Comic Book Physics, but it's sci-fi technobabble of impeccable pedigree.
Edward Elmer Smith, PhD, known to fans and friends alike as "Doc", was the author of the seminal Lensman saga, scribed back in the 1930s and 1940s. Lensman was the trope codifier for pretty much the entire genre of Space Opera, including Star Trek, Star Wars, and everything else of that nature, and had no small impact on another comic book franchise that didn't involve tiny people at all [digression] other than the occasional superintelligent alien virus.
A key piece of fictional science and technology in the Lensman saga is the conceit that later developments of relativistic theory divorced inertial mass from gravitational mass; in the epic, of course, this allowed for the faster-than-light velocities needed for star-spanning adventure.
I find myself wondering if this might prove the key to bringing some level of consistency to what the movie portrays: when Ant-Man is just standing, gravity only affects him as if he were the size and mass of an ant. When he falls, or hurtles into something, or socks someone in the jaw like a proper superhero, he has the momentum and kinetic energy of an 80-kilogram man, concentrated in the volume and surface area of one a mere centimeter tall.
(This works better if we discard the movie's explanation that the Pym Particle "reduces the distance between atoms" and return to the Official Handbook of the Marvel Universe's assertion that the mass of the shrunken person or object is "shunted into another dimension".)
It suggests that subtle variations in angle and trajectory could produce a wide range of "effective" mass and momentum. As just one example, he can match velocities with his formic steed because the ratio of his surface area to his gravitational mass allows him to control his descent, but if he jumps off, tucks in, and minimizes his surface area, he can hit that hapless mook with all the force of a full-grown man dropping a meter onto his back. (Ow.)
It's no less nonsense, but it might be the hand-wave that "fixes" the movie; I would have to watch it again with Bergenholm Physics in mind to see if it really does mesh with all the delightful, preposterous things we see on the screen [spoiler] except for that damned keychain.