My experience is that the definitions are not clearly delineated. Many research groups, conferences, etc. simply include both quantum sensing and quantum metrology in their titles, probably to attract people searching for either terminology. We also have to remember that the terms metrology and sensing have their own adherents in non-quantum communities, so using both titles can attract both types of non-quantum researchers to their quantum counterparts.
There has also been a significant amount of nomenclature discussing "quantum parameter estimation." For example, Helstrom's authoritative textbook is named "Quantum detection and estimation theory" and claims the topic to be the same as classical estimation but with density operators replacing probability distributions. This last point addressses one of your questions: density operators corresponding to classical probability distributions will not outperform "classical" estimation. In quantum estimation theory, all systems are considered to truly be quantum mechanical, so there will definitely be quantum systems and quantum estimation protocols that provide no enhancement relative to their classical counterparts.
So, without further ado, some plausible but not rigorous distinctions. In all of these, I like to keep in mind a typical estimation protocol, which can be classical or quantum: (1) probe state preparation (2) interaction such that parameters are imprinted on probe state (3) measurement of probe state (4) infering parameters from measured values.
Quantum Sensing
Focusing on creating quantum sensors; i.e., step (3) above. This means things like single-photon detectors and photon-number-resolving detectors, which classical physics is ignorant of and so would never be able to properly detect. Obviously all sensors should have quantum mechanics at their cores, but quantum sensing should involve sensors that can distinguish between different quantum mechanical states/properties.
Other things that fall into this category are super-resolving detectors, which are used, for example, to beat the Rayleigh limit in distinguishing between two point-source emitters. These detectors measure phase properties instead of just intensities, so they provide access to information that is classically unidentifiable. (You can argue that the phase information is classical if you use the correct classical wave theory, so then this is not a quantum sensor, and I have no counterargument. Both sides of the argument still agree with the premise that a quantum sensor is one that senses degrees of freedom that classical sensors do not detect.)
Does interferometry (step 2 above) count as a quantum sensor? It might seem possible classically, but not if we use input states that are not canonical coherent states. Or if we do some sophisticated interferometry technique like SU(1,1) interferometry that generates entanglement even from coherent-state inputs, shouldn't this be counted as a quantum sensor? I personally believe interferometry to fall into more than one category, which is why I think these distinctions are so hard to pinpoint. If we have a "classical" technique that allows us to take advantage of "quantum mechanical" properties, such as inputting squeezed light ("quantum") to a Michelson interferometer ("classical"), it is hard to establish whether the sensor itself is "quantum." That's why I prefer to lump it all together and call it a quantum estimation or quantum sensing or quantum metrology protocol.
Quantum metrology
If quantum sensors focus on detection, then quantum metrology should focus on probe states! In a lot of quantum estimation theory, people use things like the quantum Fisher information and quantum Cramér-Rao bound to discern the best possible result that a given quantum state could achieve for an arbitrary detector. This allows them to optimize step (1) above without regard for the ensuing steps.
This sort of procedure requires step (2) to be a given. The evolution is typically given in terms of a quantum channel, such as the unitary operations that imprint optical phases on optical field states or that rotate spins in magnetic fields. Then, one seeks probe states (1) that are most sensitive to small deviations in the parameters being applied in (2), such that the states are best at discerning those parameters. One can then compare that sensitivity to some paradigmatic classical state, which is still considered as a quantum state but one with some classical analogue, and see whether there can be some quantum enhancement in sensitivity.
The other aspect of step (1) is then how to prepare the optimal states. For example, we might know that NOON states are optimally sensitive to optical phases, but we still need to be able to generate those probe states in order to do anything useful.
Now, some more confusion. If we create an optimally-sensitive probe state like a NOON state, aren't we creating a quantum sensor? Couldn't I refer to the thing on which the parameters are imprinted as the sensor? I see no reason not to! So again, these terms have significant overlap. Also, if we combine steps (1) and (2), we get an evolved probe state on which the various parameters have been imprinted, and this is the thing that we want to be most sensitive to changes in the parameters. But this could also include the interferometers discussed above! Maybe one part of the interferometer is actually part of the probe state preparation (1), another part of the interferometer is the application of the parameters (2), and a third part of the interferometer is part of the measurement procedure (3). So interferometers play some part in multiple aspects of quantum parameter estimation, which makes categorization challenging.
Another typical part of quantum metrology is, upon finding an optimal probe state, looking for the measurements that might saturate the quantum Cramér-Rao bound. This is especially pertinent in multiparameter estimation, for which there is no guarantee that a measurement exists that can saturate the bound for all parameters simultaneously. But devising the theoretical measurement seems very close to devising the actual sensors themselves, so now we have even more overlap with my definition of quantum sensing.
Other thoughts
I believe that some people vaguely distinguish between the creation of new sensors (2-3) and the development of probe states optimally sensitive to estimating parameters (1-2), but that they are all dealing with aspects of the same overall problems.
Some people also talk about quantum imaging as a separate field, but I think it has lots of overlaps.
Definitions that specifically rely on quantum entanglement or quantum coherence are bound to raise problems because, to my knowledge, nobody has definitively identified the quantum mechanical property that is necessary and sufficient for generating some type of enhanced parameter estimation. Negativity of some specific quasiprobability distribution seems to be crucial in some contexts, and perhaps indistinguishability of quantum particles plays a role. So I would say the goal is always to measure physical parameters, to exploit some sort of property with no classical analogue, and to get better measurement precision than you can ever get with [quantum] states that do have classical analogue. This is true regardless of the title metrology vs sensing.
The types of things people measure could also lead to different titles. Perhaps sensing is more suited to things where you're not sure whether or not the item being detected exists, like radar and lidar, while metrology is better suited for measuring the values of things that you know exist, like magnetic fields or the values of fundamental constants. Perhaps this allows us to add imaging into the mix, which could cover cases like microscopy and astronomy. But of course all of these categories bleed into each other, especially because so many of them rely on interferometry to achieve their goals, so I would not put too much stock into objectively distinguishing between any of these titles.
