Q:

What is the difference between a Toeplitz matrix and a circular matrix?

A Toeplitz matrix is defined as a constant-diagonal matrix.
A circular matrix is defined as a special kind of Toeplitz matrix where each row vector is rotated one element to the right relative to the preceding row vector.
I cannot think of an example where a matrix would be Toeplitz but non-circular. 
What is the difference between a Toeplitz matrix and a circular matrix?

A:

$$\begin{bmatrix}1 & 2 \\ 0 & 1 \end{bmatrix}$$ is a Toeplitz matrix but it is not circular.
$$\begin{bmatrix}1 & 2 \\ 2 & 1 \end{bmatrix}$$ is a circular matrix.
Toeplitz matrix for a $2$ by $2$ matrix is of the form of $$\begin{bmatrix}a & b \\ c & a \end{bmatrix}$$ but a circular matrix require it to be of the form of $$\begin{bmatrix}a & b \\ b & a \end{bmatrix}$$

