You can record them if they don't abort a transaction or terminate the connection, by changing the sp to capture the value of @@error immediately after each statement in the sp, and insert it into your table if it's <> 0. You can't suppress the errors though - they will be returned to the client as well as being recorded.
Tim Wilkinson
"If it doesn't work in practice, you're using the wrong theory"
- Immanuel Kant