Here is the Wavelet transform (WT):

http://en.wikipedia.org/wiki/Wavelet_transform

Here is the Fractional Fourier transform (FRFT):

http://en.wikipedia.org/wiki/Fractional_Fourier_transform

The differences are quite many, here I sum up the most evident to me, maybe other scientists can complement:

1. The WT, at least the continuous one, is redundant, it produces at output a stack of images for a single  input image, the FRFT is not redundant

2. In the WT, each channel of the output is obtained through a convolution, therefore it is shift invariant. The FRFT is not

3. The wavelet transform is meant to achieve good localization in both frequency and time, while the FRFT is meant to represent the signal into any intermediate domain between time and frequency (in other words, at varying of the parameter alpha the output of the FRFT smoothly varies between the input signal and the classical Fourier transform).

4. The WT is scale invariant, in the sense that re-scaling the time domain results in a log-shift in the WT. The FRFT does not have this property.

I hope this helps.