What is difference between "a tad," "a bit," "a little?" Why do you use "a tad?"
Solution 1:
One would use tad when one wanted to make the expression a bit more folksy than "bit" or "little" would come across. NOAD says this about the etymology:
ORIGIN late 19th cent. (denoting a small child): origin uncertain, perhaps from tadpole . The current usage dates from the 1940
If you were writing formally you would probably use one of the other expressions. But it is certainly fair game for sports or political diatribes.
Solution 2:
I recall being exceptionally irritated back in the 80s when "a tad" suddenly became ubiquitous in the rapidly-growing context of personal computing magazines.
For several years, I never saw it anywhere else, but I used to subscribe to several such magazines, and they all used it. At the time I assumed perhaps one freelance writer who contributed to several titles just happened to like the expression.
By the 90s it was commonplace in most mass-market magazines, and for the last decade and more I've become accustomed to hearing it in speech too. 150 years ago it was just a folksy/slang term for a small child (probably from tadpole), but I've never heard it thus used in my lifetime.
To my ear, the current usage is an affectation - akin to a mite, or a smidgen/smidgeon/smidgin. It's often used where a speaker wants to distract attention from the substance of what's being said by using slightly "quirky" wording - no-one knows exactly how much a tad is, so it could be anything from "a detectable (but non-problematic) amount" to "far too much".
I'll also point out that the rise of "a tad bit" mirrors the decline of "a tidy bit". Clearly this casts "tad" as a different part of speech (adjective rather than noun), which to my mind strongly suggests we're dealing with a neologistic usage rather than something "continuous" from the original sense.