I can't tell if one of these is from 2010 and one from 2013, or if these are different types of libraries that behave differently
I have a site collection upgraded from SP2010 to SP2013. Several document library (including Style library, From Template, Site Asset, Pages) have icon same as the left one.
The right one is the new style for SP2013 document library. If I create new library in my site the icon is same as the right one.
Both of them should behave the same. At least I have upgraded our sites for over a year. However, if possible, I will create new library and move everything over. You may encounter problem in some circumstance. (For instance, export a subsite and import to another may encounter issue)