Lens Focal Length and frame size - beginner a bit confused

The more I hear and read about full-frame sensors, the more I feel they don't give any real advantage under good light conditions (and wide-angle photography, I guess, Colin ;) ). Note that I said 'sensors', not 'cameras'. Or am I missing something here?

I can see the advantages of full-frame cameras as they tend to be the top-of-the-line professional models (ergonomy, build quality, etc.), but that puts them firmly out of my financial reach... And they weigh in at a 'few' grams more as well

Remco