Molecules aren't just sums over their constituent atoms. There's many different kinds of bonds which involve different patterns in the overlap of electron orbitals, and which affect the energy levels those electrons can occupy - I'm assuming the QP video you watched explained how "color" relates to electron energy levels.
The (hydrogen-like-)atom case is just the simplest possible scenario - it's actually possible to calculate exactly. Ions/atoms with more than one electron are a bit trickier, so we can only really approximate those. The electrons in the orbitals interact not just with the nucleus (which can be approximated to be a point-charge for our scenario), but also with each other - and that's the tricky part. Now when you bond two atoms together to form a molecule, you've added another bunch of electrons to worry about; they will "deform" each other's orbitals, even those that aren't involved in the bonding directly (albeit only slightly).
For example, consider the "simple" hydrogen molecule:
While the constituent hydrogen atoms on their own would usually have simple 1s orbitals, which are "spherical", you can see that the electrons in the molecule are in orbitals that are "squashed". This of course means that the energy levels these electrons can occupy have changed, and thus also the way they interact with incident photons. And this is just a simple molecule, made of two atoms of hydrogen. It doesn't really get any easier than this. Now imagine something a bit more complex, like the rhodopshins that give us monochromatic vision - those are proteins composed of thousands of individual atoms. While only a tiny part of this protein actually participates in absorbing the incident light, this only really highlights how complicated this really is for real-world purposes.
So, why do we even care about the spectra of atoms, when we don't really find all too many "free" atoms in nature? Apart from simply understanding the physics involved, the big reason for astronomy is plasma. The surface of a star is formed out of individual ions, not molecules. So we can actually see what the star is made of by observing its spectrum. And of course, you're not limited to only using atomic spectra - you can notice that when determining e.g. the composition of Jupiter's atmosphere, we're actually looking for the "signature" of stuff like methane, or carbon dioxide - each leaves its distinct mark. The same way, we can tell the difference between hydrogen atoms and hydrogen molecules. The same way, ozone has a different absorption spectrum from simple di-oxygen molecule, which is why ozone shields us from ultra-violet light while di-oxygen doesn't.