Radial Probability Distribution Curve versus ψ² versus r curve for 1s orbitals
It is very important to note here that $R(r)$ is just the radial part of wavefunction. Wavefunction does not describe any observable. The Born interpretation says that the probability density of finding electron between any two points $x_1$ and $x_2$ is given by
$$\rho=\psi\psi^*.$$
While the probability is 
$$P =\int_{x_1}^{x_2}{\psi\psi^* \mathrm{d}x}.$$
(This is only in one dimension. In spherical coordinates you must integrate over the volume).
Why is there a difference between these two quantities? I think it will be easier to understand using an analogy. Consider a rod of unit length in which the charge varies as a function of $x$. Say the function is 
$$q(x)=x\mathrm{e}^x-1$$
The average charge of the rod is $0$. While the average charge density of this rod is $\mathrm{e}-2$. Where average is given by 
$$\langle f \rangle=\frac{\int^{b}_{a}{f \mathrm{d}x}}{b-a}.$$
I hope you have understood the difference between function density and the function itself. The radial probability and probability density are something similar. By seeing the graphs you can to a certain extent predict the average of the probability density and probability. In the second graph it is $0$ while in the second graph it is close to the peak which is $a_0$.