In a galaxy far far away... M6760 (I made it up) is an unusually massive neutron star billions of light years away and it is surrounded by a spherical Dyson sphere painted in new improved formula vantablack called Perfect Blackbody 2.0 (trademarked & copy rights) using alien tech of course (actually it is depleted carbon nanotube). So from Earth perspective, how can we tell that is it not a blackhole using modern day technology?
When we “see” a black hole or neutron star, we don’t see the actual body itself — they’re much too small, as well as black holes being black. So there’s no point messing around with Dyson spheres and black paint, because none of that will change what we see. We detect both black holes and neutron stars from their effects on the nearby matter and light, and none of those will change because of what you’re planning.
I'm surprised no one has mentioned this.
Your ultra-black Dyson sphere is easily distinguishable from a quiescent black hole or unnaturally cold neutron star by one major thing, if you're close enough to tell those two objects apart:
It blocks out too many background stars.
That is, your Dyson sphere, to avoid having surface gravity too great to support itself, will be many times larger than the neutron star, never mind the event horizon of a black hole. Whatever means you use to spot it (IR blackbody radiation, reflection at wavelengths your upgraded Vantablack doesn't absorb), by the time you can detect anything but gravity to know there's something odd, you'll easily be able to tell that, black as it is, it’s FAR larger than anything that should be that black.
Fair chance this Dyson Sphere is larger than the Golden Gate Bridge. Once you're at that size, painting it is not an event so much as an ongoing activity. You start at the Presidio, and by the time you reach the Marin Headlands, the paint where you started is at End Of Life, and you must start over.
I don't know how big Dyson Spheres are, but probably big enough to probably warrant at least 5 or 6 paint gangs. So it won't look like a black hole, it'll look more like a black melon.
We mostly identify black holes by the effects of their gravity on surrounding bodies, and the behavior of accreted matter. We have an idea of where the limit is for the mass of a Neutron Star. If we assume the obscured Neutron Star is mostly isolated from enough dust and gas for the accretion to be noticeable, we'd suspect based on the apparent mass. I suspect we'd find it very interesting, even if it is the maximum mass allowable for a Neutron Star, and would search for it to be sure we don't need to change our models. At which point, the question becomes exactly how faint the enclosure can make any radiation that tries to escape, because it is highly unlikely it could be perfectly opaque to all wavelengths.
 What happens to the enclosure if an accretion disk does form? It depends on the size and other properties of the enclosure, but my guess is that it would eventually get hot enough to have a detectable infrared presence, so I have to assume it's relatively "clean" space, and we "only" have to worry about radiation from inside heating it to a detectable glow.
The main thing to worry about is that there will be heat, and it must go somewhere. Either your enclosure will radiate it, or it will be destroyed by it. There are also other things, like polar jets etc that Black Holes give off, which it might be able to fake, but in the end, you need a way to hide the heat, and obfuscate the mass.
Given the above, you need a very massive, very cool Neutron Star, without a lot of accretable matter in the vacinity, and to design your enclosure so as to make any remaining heat difficult to detect, and to deal with any other radio signatures that would distinguish Neutron Stars from Black Holes. Deviate from any of those, and astronomers can probably tell what's up. Infrared and weird behavior of accreted matter, combined with apparently having too low a mass for a Black Hole, will get speculation going. It'd be very difficult to get just the right mass, and hide the heat, and get any necessary radio shenanigans just right, and deal with accretion realistically. The more realistic you want the disguise, the more complex your enclosure becomes, and then you have the risk of someone seeing it from just the right angle to notice that something's off when it passes in front of a star.
Mass bends space.
A large dyson sphere encircling a neutron star at the high end of the mass spectrum will bend light differently to a black hole.
The dyson sphere would have to be constructed in such a way that its mass was uniform in all directions done the planck length. Otherwise one side of the sphere will curve light more than any other part of the circle seen from earth.
Black holes, and Neutron stars produce jets of accelerated matter at their poles. The energy imparted into the jet informs us of the strength of the thing accelerating the mater. Needless to say Neutron stars accelerate mater to very different energies compared to blackholes.
The gravitational lensing creates a ring of light around the photon sphere of a black hole. A neutron star wrapped in a Dyson Sphere has less mass in larger volume, so it causes much less gravitational lensing and therefore won't create this effect. The distortion of background around it will also be much weaker.
Vantablack absorbs visible light by trapping said light in its carbon nanotubes, and that light eventually dissipates into heat, so as @dspeyer said, infrared sensors could be used to detect this change in heat/temperature whereas a black hole would not be detected by this method/observation.
If you cover the black hole with a Dyson sphere you can't tell them apart. The Dyson sphere would have to be large enough to have the accretion disk inside. You are just left with the gravitational effects. At any reasonable distance, you only see the total mass because the stuff inside the sphere will be roughly spherically symmetric.
You could tell that it is not a neutron star if the mass were too large for it to be one.