Let $$f_\pm(x):=e^x(1\pm c\sin x)$$ for some $c\in(0,1/2]$ and all real $x$.
Then $f'_\pm(x)=e^x(1\pm c(\cos x+\sin x))>0$ and $f''_\pm(x)=e^x(1\pm 2c\cos x)\ge0$ for all real $x$.
So, $f_+$ and $f_-$ are increasing differentiable convex functions that agree exactly on the countable set $\pi\mathbb Z$.