Feature return vector feature, it's like regular connecting to a database or datasource but over the web.
Vector (or as in the slide and more accuratly "Vector Tile Service") return a vector tile ie. a container where all vectorial data are cropped by the tile extend and are packaged together.
You will use the tile with a style-sheet for rendering, as the rendering is client side you could use different style-sheet with the same tile set to get the symbology that suit you the most
As a single feature may be separated between several tile this format is good for base map as you get all relevant data in vectorial format in a single tile but not so for analysis.