To the best of my knowledge here are at least two things to think about that may highlight differences between wavelets and splines.
- Some wavelets, think daubechies, do not even have explicit formula for the wavelet function and are defined by scaling functions. On the other hand splines have very explicit functions and that makes them in general easier to manipulate.
- Splines usually have the best approximation for a given function for a given order. This is when we use a spline wavelet.( Which kind of shows how related splines and wavelets are)
Thinking of concrete applications may also be helpful.
Here I generally think of wavelets as "elastic" time-frequency" localizations, for applications to video compression. I think of splines as approximation of data points , say defining a curve for spectroscopy Raman or IR and then doing some linear regression to define relationship.
Extra reading:
There is a good book by S.G. Hoogr, " Mathematics of digital Images", chapter 17, that makes the connection of B-splines to fourier transforms. And there are many papers and books that show the connection of wavelets to fourier analysis.
Hope that helps a bit.
