I'm not sure there are any direct applications of plant taxonomy, since it is mainly about naming and arranging, purely theoretical ideas. There are however practical applications of Plant Systematics. See attach picture for the main differences between plant taxonomy and plant systematics.
Here are a few examples of applications of plant systematics:
Medicine/Chemistry. If you have encountered a plant species with a certain set of interesting chemical components, and wish to use them for medicine or other chemical applications then you might also be able to find them in related species or sub-species. Related species or sub-species can be identified by plant systematics.
Plant dispersion. If you for some reason are interested in determining plant dispersion, you can investigate the biogeography of these plants. Then you can decide how far up the evolutionary tree you want to look, species, genus, family etc. for a more comprehensive picture. This can be done by identifying evolutionary relationships with plant systematics.
Plant breeding. If you are interested in breeding plants for certain physical features, chemical components, nutritional qualities etc. you can use plant systematics to figure out which species are closely related and can possibly be bred together.