I'm looking for a format for store information about data styles for my web application. There are different formats for describing vector data styles. For example, SLD, Mapnik XML, and there may be others. What are the strengths and weaknesses of these formats? What are the other alternatives?
I think the styling information format would be the least of your worries. It would be better for you to consider first what your web application requirements are. Those would then decide which platforms you would use which in turn would limit which style description format you'd end up using. This is because the style description formats are usually tied to the platforms.
For example, SLD, an OGC standard is used by Geoserver. Mapnik XML as its name suggests is used by Mapnik. Tilemill, another online cartographic tool, has its own styling format called Carto which is based on CSS. Tilemill actually has Mapnik at its core though so somewhere along the way, Carto is still converted to Mapnik UML. Mapnik web map server though. It's just used to prepare pretty map tiles for your server/web app.
I would strongly suggest that you figure out first what your web app's functions are, find a platform that can provide those functions and use the "styling formats" that come with your platform. It would be foolish to choose the styling format first and then build an application from there.