And I notice it looks like it is less than half the height of the original part.
Does that matter? Can I safely use that capacitor as long as it is rated correctly capacitance and voltage, or do I need to look up other attributes as well?
Why are some capacitors much bigger than others, yet rated similarly?
Improved manufacturing processes may allow a reduction in capacitor volume for the same capacitance (assuming it's from a reputable manufacturer, which is an unknown for that ebay item).
Those are probably okay for experimental purposes but I would buy a known brand for replacement in your power supply.
I would strongly suggest you use a capacitor from a good manufacturer, I usually stick to Rubicon or Panasonic - and using low-ESR ones is a good idea.
There's a LOT of cheap crap capacitors about, which is why they fail so much - and a PC is likely to be full of the cheapest ones they could find.
A capacitor of any size may be relatively high in capacitance and low in working voltage and vice versa. Small size capacitor implies a high dielectric constant and a high dielectric constant usually has some disadvantages in many cases.
I seem to remember something about tall thin caps being better at dissapating heat then short stumpy ones, but I can't rememberthe details...
Rated ripple current (which I suppose is related to ESR) is another thing to considder, although you might not be lucky enough to find a datasheet for your original part.