First of all, to get this out of the way, the Sun is too small to ever become a black hole or even a neutron star. The theoretical lower limit for a star to become a black hole via a stellar collapse is about 2.7 solar masses.
However, hypothetically, if the Sun suddenly turns a black hole of the same mass, there may be a potential initial burst of gravitational waves depending on the mechanism of how you intend to compress the Sun down. Other than that, gravitational changes would be minimal, if any. Specifically, the Schwarzschild (non rotating) solution does not change the gravitational field outside of the current Sun's size (as already mentioned). However, more realistic is the rotating Kerr solution. The typical speed of rotation of the event horizon is from a half the speed of light to near the speed of light. This rotation would produce a frame dragging effect. I will defer to the experts to clarify if this effect would be any different from the existing frame dragging produced by the Sun. In any case, it is likely to be negligibly small with no practical gravitational consequences for the Earth.
There would be dramatic changes in the emission spectrum. The gas around the black hole would be dragged and puled in producing a black body radiation with the soft x-ray peak plus a hard x-ray radiation from the corona. The amount of radiation would depend on the amount of gas present, but in any case the Earth would become inhabitable.