I would not phrase the question like this. It's like comparing apples to oranges. But prose aside.
Beta distribution refers to an absolutely continuous measure on a unit interval with density: $ f(x) = \frac{1}{\operatorname{B}(a,b)} x^{a-1} (1-x)^{b-1} \mathbf{1}_{0 the normalization coefficient $B(a,b)$ is defined so as to make sure that the density integrates to one: $ \operatorname{B}(a,b) = \int_0^1 x^{a-1} (1-x)^{b-1} \, \mathrm{d} x $ This normalization coefficient, as a function of parameters of the distribution, has a name, and is called Euler's beta function. Convergence of this integral imposes $a>0$ and $b>0$ restriction on parameter distribution.
The cumulative distribution function of beta distribution is: $ F(x) = \mathbb{P}(X\le x) = \frac{1}{\operatorname{B}(a,b)} \int_0^x y^{a-1}(1-y)^{b-1} \, \mathrm{d} y = \frac{\operatorname{B}_x(a,b)}{\operatorname{B}(a,b)} $ where the function in the numerator is known as incomplete beta function, and the quotient is known as regularized incomplete beta function.