Definition of Duality (opposed to Symmetry) I'm learning basic string theory right now and we came across T-duality which was presented as a symmetry of the formula for the mass of a string in the context of compactification. There was a remark that it can be shown that this is actually a symmetry of the whole theory.
So, why are we introducing a new term duality when we refer simply to a symmetry in the theory?
 A: Duality always refers to the idea that two superficially different concepts are actually, in a certain sense, equivalent. Since symmetries, rather tautologically, are equivalences, a duality often occurs as a discrete $\mathbb{Z}_2$-symmetry.
T-duality in particular, is the duality between a string theory compactified on a circle of radius $R$ with a theory compactified on the circle with radius $\frac{1}{R}$. In a certain sense, this is a duality between "IR physics" and "UV physics".
A: To help elucidate the concept of a duality, consider the Ising model in statistical mechanics. At low temperature we  can expand the partition function as,
$$Z=2e^{2N\beta J}\left(1+Ne^{-8\beta J} + 2N e^{-12\beta J} + \dots \right)$$
using diagrammatic methods. In the regime of high temperature, we may expand it as,
$$Z=2^N (\cosh \beta J)^{2N} \left( 1+N(\tanh\beta J)^4 + 2N(\tanh \beta J)^6 + \dots \right).$$
Notice both these theories are related by the exchange $e^{-2\beta J} \leftrightarrow \tanh \beta J$, known as Kramers-Wannier duality. From this example, we see a duality is a relation between two or more theories, though it is not always a concrete, rigorous concept. Tong refers to this as a 'symmetry of the partition function' under the interchange of the variables. It can be regarded as  a symmetry in that the exchange maintains the fact that the partition function is still describing an aspect of the Ising model.
For the bosonic string, the interchange of quantum numbers and radii $R \leftrightarrow \alpha'/R$ reflects the fact that the spectrum of the string is the same for both circles. The concept extends to more complicated systems. For a superconformal non-linear sigma model describing the dyanmics of a worldsheet, under certain restrictions, exchanging the target space, a Calabi-Yau manifold, for another leads to the same theory. This is known as mirror symmetry.
