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They are very different concepts. An indeterminate equation is just one with infinitely many solutions. A functional equation is an equation where the thing you want to solve for is a function, rather than a variable.
"Indeterminate equation" doesn't strike me as particularly useful terminology anyway. My guess is it originates from linear algebra, where if a system of linear equations doesn't have a unique solution, it has infinitely many solutions. But for non-linear equations it can happen that there are finitely many solutions, but more than one. For example, the equation $(x - 1)(x - 2) = 0$ has two solutions $x = 1, x = 2$.
The main difference between a functional equation and an ordinary equation is what the quantifiers are. In an ordinary equation, the claim is something like "decide whether there exists $x$ such that (some equation involving $x$)." In a functional equation, the question is "decide whether there exists a function $f$ such that (some equation involving $f$ and some other variables) for all values of the variables." For example,
$$f(x) = f(x + 1)$$
is the functional equation describing functions with period $1$. The equation is not just a collection of symbols: it comes with a quantifier requiring that the above relation hold for all values of $x$.
No, they aren't the same concept.
An indeterminate equation is an equation for which there are an infinite number of solutions; that is, there is not enough information within the equation itself to solve the equation, were it not for a "given" value at which to evaluate the function.
E.g. $y = x^2$ is satisfied by ordered pairs $(0,0),(1,1),(-2,4),(2,4)\dots (x,x^2)$.
A solution to a system of equations can also be indeterminate: e.g., the system of two equations, each in three variables, say $x$, $y$, and $z$, is indeterminate.
A functional equation is a function which is defined implicitly in terms of a function or in terms of the function at some value. For a more thorough explanation and examples, reread the links you provide in your question.
[Edited:] Seriously, as Qiaochu states, they are very different. In a functional equation, you need to solve for a function . The functional equation may very well also be an indeterminate equation, in that it admits of more than one solutions (all functions). [Thanks to Qiaochu for pointing out my earlier mis-statement of a functional equation; I've since replaced "equation" with "function"].