The video content is intrinsically the same, 480 scan lines (rows of pixels) high by about 720 pixels wide.
The "anamorphic" video was shot or telecined based on a 16:9 video frame aspect ratio and therefore looks better when the raster on the picture tube is stretched to 16:9 while the "regular" video looks better when the raster is stretched to 4:3 (or the video including any pre-recorded black bar material on top and bottom spread out over a group of pixels encompassing the respective shape).
On DVD there are flags included with each video frame so a TV with the capability automatically goes into the proper aspect ratio, or the DVD player can if you wish reformat the video as 4:3. The TV cannot tell by analyzing or taking hints from the video content.
Anamorphic DVD title has higher vertical resolution for the image since the image is stretched vertically to cover the 480 lines available in DVD format. Older non-anamorphic is letterboxed and the black bars are part of the 480 lines - its the same as letterboxed programs broadcast on regular TV channels. The DVD player when setup for a 4:3 display will letterbox (add black bars) itself to the output for anamorphic DVDs so the extra resolution is lost, but with it setup for a 16:9 display the full vertical resolution goes out.