As an information security professional, I see two different issues at play here. First, they got access. They were granted access by the admin who did not lock down the server. I am not a lawyer, but I see the unauthenticated web server, no matter how much of a mistake, as being implicit permission to access the site. A house, by default, implies privacy. A web server is more of a business in this metaphor. If the door is open and the lights are on, it's implied you can come in and look around. Machines accessible over the web are by default open to everyone unless permission is revoked. The "unauthorized access" charge, in my opinion, should be struck down. When a site is made accessible from the unauthenticated Internet an admin implicitly granting you permission to visit the site.
The second issue at play is the fact that the guy apparently collected some email conversations to use as proof. Using my business metaphor, walking into a closed business that to a layman appears open is a simple mistake. Anyone could reasonably assume the business is open. However, collecting their merchandise even just to prove they forgot to lock up would still be stealing. In this situation, it's unauthorized copying. Most reasonable people would consider this to be unacceptable.
The second situation is muddied a bit further by my wording "most". Websites accessible when unauthenticated are able to be scraped easily. What if the Googlebot crawled the site and collected the information due to a poor robots.txt? What if you walked into the business and tried some free samples (unauthenticated websites are implicitly free samples)? Data privacy comes into play on this one though, and I would argue that any reasonable person would understand these as private communications. While they are accessible to view, any reasonable person would understand it is unethical to read them and unacceptable to copy them.
The fatal flaw of the defendant was copying the emails. Up to that point, he was completely within reasonable practice in my opinion. Here's a takeaway for any startups: security isn't a joke. It's a career ender, it's a business ender, and it could be a career ender for your customers who trusted you. You hire the best programmers, but budget a little aside for an external penetration test, and take the results seriously. Don't lose your company and your reputation because of a caffeine-fueled oversight.