Hi dms,
Both options you listed are excellent choices as I'm sure you found out through reviews. KeePass is known for being very in-depth accompanied by a great knowledge base online, however, it can be less user friendly for first time users, and you are responsible for all syncing between systems (e.g. pairing with dropbox)
LastPass on the other hand is known for being one of the most convenient options available, easily being able to sync with the cloud and other systems and having a good mobile option. Since it is browser based, you won't have any problems working between operating systems. However, the mobile version is not free and requires a premium subscription; fortunately, it's only a buck a month ($12 annually). And even though you do sync with their servers, everything is done locally on your machine and encrypted.
At the end of the day it'll come down to what you find to be the most appropriate for your daily needs. I personally find LastPass to be the better option as it allows for more flexibility in terms of convenience (synchronization, easy auto-fill, easy organization, cross-platform, etc). I also use two-factor authentication which I would recommend everyone turn on if they use LastPass as @
LookinAround points out. So I have a password (something I know) and an authentication key (something I have).
Here are some more in-depth links that will give you a better idea about LastPass.
Which Password Manager is Most Secure?
Security Now – LastPass Security - (LastPass discussion starts around 52:45
Also note that whatever choice you make, you
MUST choose a strong master password. The whole point of having a password manager is so you only have ONE master password to remember. If you work with a lot of sensitive information, it might not be smart to store everything under one account as well.
The below link should give you a good idea on how to craft a strong master password.
Schneier on Security - Choosing Secure Passwords
One thing I still do not understand is, whichever one I have, what happens when I visit my mother -- how do I use HER computer to access my bank account?
I would recommend against using anyone else's computer to access your accounts unless you explicitly know it to be secure. As much as I love my friends and family, their computers may not be secure.
But if you must, you can simply access your LastPass vault online, or install the LastPass extension/program for KeyPass and sync up your account and delete it when done. Even better, use the mobile versions of the password manager on your phone, and grab the password you need, just don’t save it on their PC. You're still vulnerable as pointed out above, but at least only one account is and not everything if you don't use two-factor authentication.
Hope this gives you a better idea on what to choose.