I personally find 0 cleaner than 0px. That's two extra characters that can add up. Why add extra bytes when you don't need to. I have see padding: 0px 0px 0px 0px which can easily be expressed as padding: 0 way too often.
As far as I'm aware there is no difference between them, since 0px = 0em = 0ex = 0% = 0. It's purely up to you, as the developer, to decide what you like best (unless you have corporate coding standards that you need to follow, of course).
From most of the code samples I've seen, most people use the unitless version. To me, it just looks cleaner. If you're pushing a significant amount of data (say, if you're Google), those two bytes can add up to a lot of bandwidth, especially since you're quite likely to repeat them multiple times in your stylesheet.
As the others say, it doesn't really matter if its 0, though I choose to add the measurements to all of my values so anyone else looking at my CSS files can gauge what measurements they're likely to deal with elsewhere.
The format of a length value (denoted by <length> in this specification) is a <number> (with or without a decimal point) immediately followed by a unit identifier (e.g., px, em, etc.). After a zero length, the unit identifier is optional.
While the unit is optional when the value is 0, I tend to leave it in, as I can then tweak the values with Chrome's Developer Tools by clicking on the value and pressing the up/down arrow keys. Without a unit, that isn't really possible.
Also, CSS minifiers strip the units off of 0 values anyways, so it won't really matter in the end.
You can use either - my best advice is not to worry too much but be consistent in doing it either one way or the other. I personally prefer to specify '0px' for the following reasons:
Using 0px makes things more consistent with all of the other 'px' sizes you've specified
It's also more verbose and makes it very clear that you're setting a zero length rather than a 'switch this off' flag
It's slightly easier to tweak a '0px' value to make it another value if required
zero-units Zero values don't need units. An easy way to save bytes in CSS is not include units when a value is 0. For instance, 0px and 0 are the exact same ...
If somebody gives you 0 EUR. It is that same like 0 Dollar or 0 Zloty etc. What you got is nothing = 0. That is why in the case of 0 you dont need to set a unit.