As with much in the English language, it's entirely dependent on context and tone. I would say they are different but not actually in the difference between the definitions of 'little' and 'big'. Best illustrated with a few examples.
If my daughter said "Daddy, the mice ate all the chocolate biscuits again" with a naughty grin on her face, I might say either of those in reply with a grin and it'd be completely benign. Clearly she didn't intend me to believe what she said (that was a running joke between us when one of us had eaten something the other might have liked to have had some of).
In the benign case, the 'little' would be implying that it was a small lie and/or that she was in the habit of telling little lies, perhaps better known as 'fibs', 'little fibber' would be a better phrase as 'fib' implies a non-serious lie and makes it clear to the listener that you don't see this as a serious matter. The implication that she is in the habit of telling little lies itself is a benign one, and not meant to be taken seriously either.
In the benign case, the 'little' doesn't really refer to her being a child or physically small as one wouldn't change it in any way if saying it to one's spouse, for example.
If I said the same "You little liar" without the grin, and a more aggressive or even a fairly neutral tone she would immediately counter with "No Daddy, I didn't really mean it, I ate them, sorry" or similar (or just start crying I guess !), the point being that she would instantly know that I had mistaken her intent to make a joke which she didn't intend I take seriously, even though the words I said were exactly the same.
I would say that 'big' in a friendly context meant exactly the same, that it was a small lie, but British people commonly use 'small' to imply 'large' and vice-versa, it's all in the context and how it's said.
There is some element of familiarity (whether good or bad) implied by the use of either, "big liar" it's not generally a phrase an adult would use to another adult.
If said in reply to a work colleague saying "I didn't touch your wallet", when you just saw them putting it back in your jacket one might use either, but 'little' would be far more usual. In that case the 'little' would be a diminutive slight aimed at them. Not at their physical size or age, but purely as a demeaning slight.
Adding the 'little' in a negative sense is always about adding a barb to the statement, never about the lie being a small one.
One would be unlikely to say "You big liar" to a stranger, as it lacks impact and just sounds childish, it's not something you'd really say to seriously confront somebody whether you knew them or not. You'd simply say "You liar" if you were going to be terse about it or the rather more reserved British "I think you are lying", which might be said in an entirely level, calm manner but have more aplomb as a result.
The 'little' said to a woman doesn't make it specifically sexist as claimed above, it's demeaning to anybody regardless of their genetics when used in a negative context unless the definition of sexist means "Anything negative said to somebody of the opposite sex for any reason", which clearly it isn't.
In all cases, the implication of the big/little is found in the context of the exchange and the vocal inflection and any physical expressions when it is spoken.
I'm a mature well-read native English speaker with a classical education.