The only real difference between a blaack hole and a star, gravity-wise, is that you can get much closer to a black hole. How close can you get to the Sun? Well, you can get to the surface, but once you reach the surface, some of the mass is above you, and there are a bunch of non-gravitational forces mucking things up. A stellar-mass black hole, though, you can get within a few kilometers of the center.
Black holes can, in theory, be (almost) any mass (there might be some difficulties below about the Planck mass or so), but there’s no known way to produce them in the present Universe smaller than a medium-large star. Once you have one, it’s very easy to make it bigger, but the only way to make it smaller is to wait a very long time for the Hawking radiation. There may have been smaller black holes produced in the Big Bang, and some of them may still be around, but they’ve never been detected.
By the way, nobody’s mentioned this yet, but a black hole moving towards us isn’t necessarily anything to worry about. Radial velocity (whether something is moving towards us or away) is much easier to measure than transverse velocity (side to side), so when astronomers report that something’s coming towards us, it’s probably coming sort of past us at an angle, not directly towards us.