The DVI is digital. The trapezoidal is analog. The LCD itself is a digital device.
The video card itself is a digital device.
The analog output converts from digital to analog (on the card) and sends the signal out.
The analog in on the LCD takes analog signals and converts them to digital (on the monitor) to get the proper signal.
So, basically you're wasting time going from digital (card) to analog (cable) back to digital again (LCD) and possibly losing quality along the way (?).
The card does it in case you have an out-dated monitor or CRT, which use analog.
The monitor does it in case you have an out-dated card that only supports analog.
Both are legacy support for the older format (analog).
I've heard the DVI can transfer more data at faster rates, but I think that was in general, and that for monitors you won't notice MUCH difference (IMO you can probably see a little).