I think I've figured out why the server stats spiked with the new server.
After looking at the web server stats and access logs, the visit volume hasn't changed, but the bandwidth per visit doubled for some reason. While monitoring the access logs, I noticed that page requests were being logged twice for some reason. So either the webserver is logging things twice for each request, or the server is serving up each request twice, effectively doubling the bandwidth.
So, now to figure out why Apache is acting up like this and how to fix it.
Update: Found the culprit. Turned out to be an extra CustomLog directive in the config file that was causing requests to be logged twice. That means the increased web server bandwidth is just an illusion.