DVI and VGA were at one point the most common methods of connecting monitors to devices, but decades later, what's the practical difference between them?

I need to go from a laptop that has VGA only, to my HDTV which has everything except VGA. I've tried the VGA-Component(RGB) adapter cables and that did not work. I also tried the VGA/DVI adapter and ...

I have a 17" monitor with a VGA cable that seems to have gone bad or is losing its connection at the monitor end. I think some of the pins or wires inside are not getting contact or got detached, ...