What one really can do with fractals built from L-systems? For any L-system one can naturally associate a fractal. Why these fractals are (mathematically) useful apart that they are a source of nice pictures?
 A: The original use of L-systems is to model the fractal-like forms that appear in plant growth. Johan Knutzen gives a nice overview with pictures in Generating Climbing Plants Using L-Systems.
In computer science, L-systems are used to generate the space-filling fractal curve that maps IP addresses to computers (Hilbert curve).
More recent applications include musical compositions with a self-similar structure, as described by Stellios Manousakis in Musical L-systems. (This musical rendering of L-sytems has been called Growing Music.)
------ Here is how an L-system fractal sounds in C minor.
More musical compositions based on L-systems can be found here.
A: The Wikipedia page for L-systems links to the Thue-Morse sequence for which there are a large number of applications.  In particular it can be interpreted as the sequence of distances between atoms in a 1D quasicrystal.  Hence I suppose you might think about the work on substitution tilings and higher dimensional quasicrystals as an application of similar ideas.
Thinking even more broadly, the recent work by Borrelli, Jabrane, Lazarus and Thibert on implementing convex integration for a $C^1$ embedding of a flat torus in $R^3$ uses explicitly an L-system-like rewriting process to add corrugations.
