I think there is a place for each of the three basic forms of logs. I tried to give a brief overview but it did get a little long.
1 - Good old pen and paper
Pluses
[ul]
[li]It’s cheap, reliable and easy to make drawings.
[/li][li]Doesn’t require batteries or internet access.
[/li][li]Less duplication of information.
[/li][li]There’s just something personal about a nice hand written log that is lost in technology.
[/li][li]Plus if you spill that beer on your paper log it adds character. If you spill it on your laptop 
[/li][/ul]
Minuses
[ul]
[li]Not as easy to integrate pictures
[/li][li]not as easy to share information,
[/li][li]not as easily searchable. Especially as the log gets larger or with multiple logs.
[/li][/ul]
2 - Locally on a PC/Laptop/Blackberry/PDA logs (access, excel, OneNote, specific fishing log applications, …)
Pluses
[ul]
[li]you don’t need wifi/internet access.
[/li][li]You can keep your laptop in your vehicle for mid-day notes.
[/li][li]Easy to email/print information for others.
[/li][/ul]
Minuses
[ul]
[li]Electronic devices are theft items.
[/li][li]Many run on batteries.
[/li][li]It’s something else to lug along on a fishing trip.
[/li][li]Since you don’t usually take these to the stream you may need to record notes along the stream and then key them in later.
[/li][/ul]
3 - Online logs (there are quite a few to choose from)-
Pluses
[ul]
[li]Offer many of the same benefits as locally stored logs.
[/li][li]Some additional benefits are you can read/write notes from any internet accessible device. So if you don’t have your laptop/Blackberry/whatever you can still make notes using someone else’s device.
[/li][li]This method is also the easiest to share information. You simply open your log for others to see or send them a link.
[/li][/ul]
Minuses
[ul]
[li]Internet/wifi is required otherwise you’re writing it down somewhere then re-entering the information online later.
[/li][li]You need to see if you can make local backups that can be read by application on your laptop. Otherwise, if the site goes out of business so does your log.
[/li][/ul]
Some people use more than one method. I tend to use option 2 since it seems to fit my personal preferences and I can’t read my own writing at times.
I use OneNote since it’s easy to include rough drawings, add links, scan in articles or streamside notes, attach documents, copy web pages, include photos,… Plus it’s very easy to create your own stationary that only includes information you feel is important. I also use the same application for hunting logs & trip planning. I haven’t found any application that’s this flexible.
If anyone is interested in how to use OneNote let me know.