SHP is several files, limits field names to 10 characters and is not web-friendly. WKT is what spatial databases and GIS libraries accept as text. SHP is at home in exchanging vector map layers between GIS programs; WKT is the usual choice for writing a single geometry in a database or script.
Every format is read into plain GeoJSON features and written out from there, so coordinates, names and properties carry over wherever the target has a place for them. A zipped shapefile (the .shp with its .dbf and .shx) or a bare .shp is read. If a .prj file names the coordinate system, coordinates are converted to latitude and longitude; a coordinate system that cannot be read is refused. WKT describes shapes only, so names and properties are left out; each feature is written on its own line. Coordinates outside the latitude and longitude range are refused, since the file is probably in a projected coordinate system.