When I first started to learn to play around 2005 I only used synthesizer semi weighted keyboards. I quickly got accustomed to that feel as I grew more comfortable with my playing and the speed. The shock came was when I tried a fully weighted keyboard at a friends place. I realized that I could not play as well as I could at home with my light semi-weighted keyboard. I was told that this was a common problem with keyboardists that never really learned to play a real piano. I started practicing at some local music instrument stores on my way home and also with band practice in some studios.
Not only did I gradually become used to the weighted keys, I started to feel the advantage. To me, the advantage seemed to be the kick back motion of the keys helped with the rhythm of the playing, especially faster tempo melodies i.e. jazz. You would think the opposite but that bounciness really facilitates your rhythm. I noticed the difference when I went back to playing my semi-weighted keyboard.
Another advantage of a weighted keyboard is how well you can play quietly. Semi-weighted keys are of course touch / velocity sensitive however it is not too easy to play very quietly. You may need to adjust the velocity touch settings.
The only disadvantage I can really see with weighted keys is the actual weight of the whole keyboard and the cost. Semi-weighted keys even 88 keys are very portable and perfect for travelling gigs. A 88 key fully weighted keyboard is extremely heavy and definitely not portable. Also, if you are only used to semi-weighted keys then your fingers can get pretty tired after lengthy playing.