What is the difference between a functional and an operator?
Loosely, an operator (acting on a function space) takes functions to functions (e.g., $f(x)$ to $-i f'(x)$). On the other hand, a functional takes functions to numbers (think about a certain integral, or the derivative evaluated at a certain point).
An operator is a (not necessarily linear) map from one vector$^1$ space or module to another.
In operator theory, it is usually implicitly assumed that operators are linear.
In quantum mechanics, it is usually implicitly assumed that operators are linear or antilinear. (However, see Wigner's theorem!)
A functional is a (not necessarily linear) map from a vector$^1$ space into a field.
In the topics of calculus of variations and Lagrangian mechanics, the functionals are typically non-linear.
In functional analysis, it is usually implicitly assumed that functionals are linear.
--
$^1$ This is Wikipedia's definition (October 2016). However, since the map is not necessarily linear, there is more generally no reason to insist on vector space structure in the first place. E.g. physicists would call the WZW action a functional, even if its domain is technically not a vector space.
Mathematically, we have lots of words that all refer to the same general idea — the precise meaning of a word is not one of universal definition, but of linguistic convention that develops in various subjects (and even then isn't always consistent).
That said, in my experience (as a mathematician), in linear algebra contexts, "functional" is nearly always reserved for scalar-valued linear functions, and "operator" is usually used for an element of some sort of algebra when one intends to work with representations of that algebra (e.g. the algebra of linear endomorphisms of the space of complex-valued functions on the reals, with its representation of acting on said space of functions).
The use of "functional" as meaning any sort of function whose domain includes functions tends to happen more in domains like formal logic or computer science.