Conceptually, what is the difference between the Beta function and the Beta distribution? I have read the Wikipedia pages on the Beta function and the Beta distribution, but I'm still not sure I have a good intuition for what's going on.
I'm am hoping someone will be kind enough to explain in an understandable fashion what the two are doing differently, and how to think about them.
And-- is there any way to use the nice easy formula for the expectation of the Beta distribution to get an evaluated form of the Beta function?  
 A: 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:
$$ 
   f(x) = \frac{1}{\operatorname{B}(a,b)} x^{a-1} (1-x)^{b-1} \mathbf{1}_{0 <x<1}
$$
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.
