Although I doubt this will have a satisfying answer, I still think it's an interesting question. Complex Analysis is precisely interesting because differentiability on a small neighborhood implies analyticity, which (in a certain sense) means that information at a point propagates to nontrivial local information. So investigating the real-analog is a terrific thing to be curious about and makes you appreciate Complex Analysis much more.
There are many ways in which Real Analysis is "ugly," and your question can exemplify one way in which this is the case.
Considering the difference of any two functions you're working with, your question is equivalent to asking
On an interval, just how "zero" can a function be without being equivalently zero?
Exploiting graph transformations we can focus on the case where $f:(-1, 1)\rightarrow \mathbb{R}$ is $C^\infty$.
Of course, if $f$ is zero on a dense subset of $(-1,1)$, then $f$ must be equivalently zero. This only exploits continuity.
In hopes of constructing a weird function off a weird subset, this might lead you to ask, "just how dense can something be without being dense?" The answer is "not much". If a subset is not dense, then there will be a point and a neighborhood of that point where the original wannabee-dense subset does not even have a single point in common with that neighborhood. In short, a subset that isn't dense doesn't show up in SOME open subset.
Turning back to the original problem here, a function that isn't zero on a neighborhood is pretty much not zero in my book.
But you might be comfortable with a function not being zero on a dense subset and still consider it "pretty much zero". Ways in which I can think of this being reasonable is a function which is zero on a closed nowhere dense set which has positive Lebesgue measure---e.g. a "fat" Cantor set. These are examples of subsets which are "small" topologically but "big" measure-theoretically. You can't quite construct a nowhere dense subset with full measure. You can get a full measure subset with empty interior, but this subset will be dense (and thus not interesting here).
For any closed set $F$, you can construct a function $f$ which is $C^\infty$ on $\mathbb{R}$ such that $f^{(n)}(x)=0$ for every $x$ in $F$ and any natural number $n$. This is one of the weird things about Real Analysis. Using this construction on something like a fat Cantor set would give you a pretty-much-zero function. So one half-way of answering your question is how would you tell such a function like this from the zero function? You would certainly need to know this function's value at some other point. The places where it is not zero is dense and open; however, you might have an arbitrarily small probability of picking a point where it evaluates to something else. Quite the conundrum. Depending on how "fat" this Cantor set was, the universe might die out before you have an appreciable probability of picking a point at random where it evaluates to not zero.
It seems you'd need to know how the function behaves on an entire interval, which isn't very satisfying.
