The integral closure of a complete Noetherian local domain is also complete, local and Noetherian. So the integral closure of $R/p$ is a complete regular local ring containing a field (normal rings in dimension $1$ are regular by Serre criterion), and you win!
EDIT: add more details, by Tymothy's comments (now deleted): So let $S$ be the integral closure of $R/p$ in its quotient field. Then by definition, $S$ is integrally closed (normal), thus regular because $\dim S=1$. Clearly $R/p$ is a subring of $S$. By Cohen Structure theorem, $S$ is a power series ring with one variable.
The fact about integral closure in my first sentence is well-known. Check out Theorem 4.3.4 of the book available online by Huneke-Swanson, or Eisenbud Chapter 13, or Matsumura somewhere (probably the finiteness of integral closure is true for excellent rings as well).
The best way to see this: forget about the $X$ in a polynomial or a formal power series, they are really sequences of coefficients, with no constraint on their values, with some specific rules of computation for addition and multiplication.
A power series, however, involves some limiting process, and that requires specific conditions, namely that the series converges.
For instance, you can manipulate $S=\sum_{n=0}^{\infty} x^n$ as a formal power series, and you won't consider convergence, only the operations on it, for instance $S^2=1+2x+3x^2+\dots$. That is, the coefficients of $S$ are $(1,1,1,\dots)$ while the coefficients of $S^2=(1,2,3,\dots)$. But you may also consider $T=\sum_{n=0}^{\infty} n! x^n$, it's a valid formal power series.
Now, for a power series, you require convergence. It's possible to prove that a power series in $x$ converges for all complex number $x$ such that $|x|<R$, for some real (of infinite) $R$. This $R$ is unique and is called the radius of convergence. For instance, the series $S$ above has radius $1$. It converges for $|x|<1$ to the number $\frac{1}{1-x}$. The series $T$ has radius $0$: it never converges if $x\ne0$. As a power series, it's almost useless, but as a formal power series, it can still be useful (we don't care that it does not converge).
There is a similar distinction between a polynom and a polynomial function. But here it's even more tricky, because in the usual undergraduate courses polynom are considered with coefficients in $\Bbb R$ or $\Bbb C$, and there many properties of polynomial directly relate to properties of the associated polynomial function.
When coefficients are in a finite field, it's more surprising. For instance, in $\Bbb F_2$, the finite field with two elements, the polynom $X^2+X$ is not the null polynomial (the null polynomial has null coefficients). However, the function $x\to x^2+x$ only takes the value $0$.
Best Answer
Well, an elementary but striking difference is that the polynomial $1 -X$ has no inverse in $k[X]$, but has an inverse in $k[[X]]$, namely the series $\sum_{n \geqslant 0} X^n = 1 + X + X^2 + \cdots$