Dedicated lines are exactly as secure as the internet - i.e., not secure, so there's no difference on that regard. From a security viewpoint, they're treated as unsecure as a number of third parties (the network providers and such) may have access to it; so you have to encrypt all data in transit in the exact same manner as you would on an internet connection.

In finance, you get dedicated lines for bandwidth/latency guarantees (i.e., no congestion out of your control) or for redundancy and fault tolerance; not for security.