Discretized CWT vs. DWT vs. FWT vs. MRA
For the CWT the values in your definition can be any $a\in \mathbb R^+$ and $b\in \mathbb R$ if applied to functions. The CWT can also be applied to discrete signals (discretized CWT) with $b$ restricted to appropriate discrete values (and any $a$, but typically $2^{j/v}$ where $v$ is the "voices/octave" of the transformation). Now when we restrict $a$ to some discrete values $2^j$ we get the DWT, which is an orthogonal transformation.

(Small edit to correct the scale : $2^{j/v}$ instead of $a^{j/v}$)