Most major tournaments (including the World Championship) are commonly played on DGT Boards which automatically record the notation for the players. This has some distinct advantages;
- Improved ability for broadcasting
- Reduces need to transcribe games after the fact into a digital format.
That being said I believe it is still a requirement for players to manually record there games. I don't believe these records are discarded as they are submitted to the tournament organisers who technically own them.
In regards to why pen and paper. It minimises opportunities and accusations of cheating. If I was playing over the board to you and was entering in each move onto a computer (through something like chess base) who is to say I didn't enable an engine to quickly analyse a move?