It allows you to store the information of a signal in a small number of coefficients.

in wavelet transform only approximate coeffitients are further decoposed into uniform frequency subbands while in that of wavelet packet transform both approximate and detailed coeffitients are deomposed further into sub bands.

Fourier transform analyzes signals in the frequency domain, representing the signal as a sum of sinusoidal functions. Wavelet transform decomposes signals into different frequency components using wavelet functions that are localized in time and frequency, allowing for analysis of both high and low frequencies simultaneously. Wavelet transform is more suitable than Fourier transform for analyzing non-stationary signals with localized features.

cousine transform orthogonality is good

They are similar. In many problems, both methods can be used. You can view Fourier transform is the Laplace transform on the circle, that is |z|=1. When you do Fourier transform, you don't need to worry about the convergence region. However, you need to find the convergence region for each Laplace transform.

The discrete version of Fourier transform is discrete Fourier transform, and the discrete version of Laplace transform is Z-transform.