There are a few ways, some simple and some complicated, but above all you will just get a grasp of the object by playing the game a lot.
1) You will be dropped out of hyperspace much earlier in a white dwarf system than at a neutron star system. That's because white dwarfs have much larger exclusion zones than neutron stars. Thus, if you finish your jump really close to the star (the tips of the jet cones are almost 180 degrees apart on your screen) that means it's probably a neutron star. If you can fit the whole jet cone in your FOV (roughly), then it's probably a white dwarf because you got dropped out earlier. You can also look at the distances the star is from you if you don't want to judge using stadiometric methods. I don't remember exactly the distances but I think neutron stars will drop you within 1 ls, whereas white dwarfs will drop you a few ls out. A more experienced player will be able to judge the distance by the handling of their ship in supercruise (the sound of your ship overspeeding in the gravity well is very apparent in a white dwarf well) as well as notice that the white dwarf doesn't move much relative to you going at the speed cap of 0.33c near a star, whereas near a neutron star if you do that you will zoom past in a second.
2) The shapes of the jet cones. Neutron stars rotate faster and have a roughly cone shaped jet cone. Even the slowest rotating ones will still rotate fast enough for you to see. White dwarf jet cones look more like splayed wires and I have never noticed them rotate that much.
3) On low graphics settings, white dwarfs have a grey circle around them. Neutron stars don't. But at that point there is no lens flare so...
4) White dwarfs are much hotter and you will overheat very easily near them. This is useful if your ship isn't the coolest running.
5) Look at what you're jumping to before you actually jump to it. I make it a habit to look in the top right of my screen and see what I'm jumping to, no matter what. This will save you when you run into a white dwarf (you will be prepared) and can also reduce the risk of running out of fuel because you will double check your fuel when jumping to a non-scoopable.
There's probably a few more reasons but these should be sufficient.