Many people don't log dives,
Some do digital logging of dives and,
Some do paper logs.
In the dive club I belong to I would say it is 50/50% "loggers" and "no-loggers",
and of those who log dives 50% logs dives on paper the rest do it on a PC/online.
I log dives both on paper and on my PC (the paper log is handy when I am on the boat, or traveling), the digital log contains more info like dive profile, theoretical tissue saturation, stats and so on.
Last time I did scuba in Thailand most people logged their dives on paper, there was some old-timers that didn't log their dives.
I guess the dive log is most useful when you start out as a diver (let say the first 50 dives) to make notes on what worked and what didn't. as you get more dives under your belt stuff like weighting and gear configuration is something you just know, and don't need a log to remember, that said I know a woman that have been an instructor longer than I have been alive, and she still do paper dive-logs.
I log my dives because it have become a ritual, sitting down with a cup of coffee and the logbook and talking about todays dives, and also sometimes it is nice to just look at the logs and think back on those special dives. The digital log is also nice to link images taken on dives to place, date, time, depth, weather condition and so on.