As stated, Heller protects the ownership and use of firearms "in common use for lawful purposes", but the context was handguns.
The category of handguns is massive in the firearms world. If one starts at the very top trying to categorize firearms, the first logical division is between long guns and handguns.
So banning the possession and use of handguns (as was the issue in Heller) would be a huge impairment of the rights of gun owners. It basically removes one of the two major classes of firearms and eliminates, or at least severely handicaps all of the intended uses that were the reasons that handguns came into existence. The differences between handguns and a long guns are very significant. There are things you can do with a handgun that simply aren't possible with long guns.
Does banning semi-automatic centerfire rifles with certain characteristics rise to the same level of impairment? Let's see.
Looking at the firearm classes and thinking about how a law might be written to ban "assault weapons", we would start with two major divisions of firearms:
Handguns and long guns.
Long guns could be divided into rifles and shotguns.
Rifles could be sub-divided into centerfire rifles and rimfire rifles.
Centerfire rifles would be subdivided into repeaters, double rifles and single shots.
Centerfire repeating rifles would be split into manual repeaters, semi-automatics and fully automatics.
Semi-automatic centerfire repeating rifles would break into subcategories based on the features that separate "assault rifles" as defined by law (detachable magazines, pistol grips, etc.) and those with only the features that are considered to be benign.
So the class of firearms being banned would be:
Semi-automatic, centerfire, repeating long guns that are rifles with certain features that meet the law's definition of "assault rifle".
Now, there's no debating that category is in common use for lawful purposes, but it's equally obvious that it is a much smaller subdivision of the entire "world" of firearms than the category of "handguns".
What about levels of ownership? Studies indicate that around 3/4 (72%) of gun owners have a handgun. Among gun owners, less than 2/3 (62%) own rifles. So we can already see, before we even start subdividing the rifle category into centerfires, semi-automatics, those with detachable magazines, etc. that there are fewer gun owners with rifles than with handguns. It's pretty obvious that the ownership of "assault weapons" is going to be much smaller than the ownership of rifles--even much smaller than the category of centerfire rifles.
For whatever it's worth, this is likely, at least to some extent, by design. Gun banners know that they have to take small bites or they will get slapped down. So attacking just one small category of gun owners at a time is a necessity if one wants to make progress at banning firearms.
Does banning "Semi-automatic, centerfire, repeating long guns that are rifles with certain features that meet the law's definition of 'assault rifle'" fall into the same level of rights impairment as banning one of the two main categorizations of firearms? I think it's clear that it does not.
The question is whether or not SCOTUS believes that it is sufficient impairment to justify striking down the law and I honestly have no idea how to predict that. At least one appellate court has ruled that it is not sufficient impairment and that assault weapons can be banned under Heller.
http://guptawessler.com/wp-content/uploads/2012/05/Kolbe-v.-Hogan-en-banc-opinion.pdf