Runtime errors can basically occur for two reasons.
1) Something completely out of your control during implementation
Attempting to read from a socket that got closed, trying to read a file that was deleted from disk, or a set of dependant libraries being removed can all cause runtime errors.
2) Not properly checking input from users.
One of the classic examples is when doing database work. If you forget to properly translate strings passed to your code (or you don't use the right functions), and a user inputs the string:
Bob's Birthday
The single quote can totally screw up your SQL statement.
Logic errors, on the other hand, are much more dangerous (in my opinion). This is because a logic error can sometimes go unnoticed for a long time. If a user is using the system to input data, and the data gets manipulated and stored, it's entirely possible that a logic error is causing the data to be stored incorrectly. As far as the user can tell, the data gets into the system, because no big error box tells them otherwise. This can cause serious latent issues in stored data.
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