Are there things an analog computer can do that digital computing cannot do?
Digital computers are also analog if you get down all the abstraction levels until you reach electrical circuits. The only difference is that we choose to "cut" as with some sort of grid in what levels of detectable analog shifts in signal we create another new abstraction level that we call a bit or a byte.

Anything an analog computer computes, like for example the output signal of an analog filter or the amount of millimeters that a mass in a spring and damper system moves, will also eventually reach a maximum resolution. This is because of for example noise in detectors, error in measuring equipment, and maybe quantum phenomena though I'm not so sure as how that would work. If you define a bit as that very small quantum of information you'll get a digital abstraction for the output of your analog computer.

In other words, if you use a formal abstraction of a digital computer where you can achieve enough resolution for your computations, you'll be able to compute the same than an equivalent real-world analog computer for that problem will compute.

The same problem occurs when digitizing time. For example if you look into analog and digital filter equivalencies, there is always an error that is introduced when digitizing an analog filter. That error tends to 0 as the time step used for the discretizing of the continuous system gets smaller.