If your jar is in common then it's loaded globally in tomcat. Every webapp sees it. If you put it in your webapp only your webapp sees it. What I assume Boris is referring to is if some part of your global tomcat config loads something that needs a mysql connection (like a connection pool) then it's going to need the jdbc driver, so it'll need it in common. Otherwise, no.
I believe there's a security configuration you can set up in the tomcat config that stores its config in a db. If you use this, it'll need a driver.
Why would you want stuff in webapp/WEB-INF/lib? well, because it's modular, it's part of your webapp, if you move your webapp somewhere else it doesn't need an extra part of it that's part of your tomcat installation. Maintenance is much easier if you can drop your webapp directly in a stock tomcat installation. Another reason- if you have multiple webapps, they're all going to use the jars in common, which could cause library conflicts and version issues.
In general, put as little in common as you can get away with. Even if you only have a single app.