The obvious big thing you miss out on is a tour guide.
But you and I think very much alike. When traveling, I like to save on paid tours by using audio tour podcasts (or podwalks), so I immediately checked my various audio tour apps and podcast bookmarks, and you are in luck! You are going to the right place, because Copenhagen X actually claims to have invented the term "podwalk".
There is an excellent multi-part Harbor Bus Copenhagen tour available in podwalk form from the Danish Architecture Center. I'd also check out the Your Harbor introduction to the harbor and the new bridges connecting around it.
You can download all the podwalks from the Center's website, but they are also on iTunes, which might make getting them onto your phone or media player a little easier.
At poster's request, here are a few words about how to handle the things you might be missing vs. a paid canal boat tour. My answer is more philosophical than technical: Simply do not worry about it.
Complement the harbor bus ride with a walking tour such as the Sandemans New Europe free tour, which I have found to be of good quality in the many European cities where I have taken it - a great way to orient yourself to a city. Then trust that if there is anything else truly unmissable, a local or a fellow traveler on the walking tour will soon let you in on it. That usually happens wherever I go. It is difficult not to hear about the best places - in Stockholm, even the locals repeatedly praise and recommend the Vasa Museum; in Riga, it was the gorgeous Art Nouveau enclave "The Quiet Center" around Elizabetes Street.
It would certainly be possible to trace every site visited by a typical canal boat route and match it to the Havnebus map to make a strategic plan to fill in every gap. But often this kind of military-grade planning disrupts the opportunity for serendipitous coincidences. My experience has been that if you leave a few gaps in your plan, they will invariably be filled by richer adventures than you would have had if you micromanaged every detail.
So my recommendation is this: take one of those little battery-powered phone boomboxes with you. Take the walking tour, and make a few friends along the way. Invite them to ride along on your Havnebus tour and listen to the podcasts on the boombox. Head to the pub afterward with your new friends, and ask them "Has anyone seen anything in Copenhagen that they consider unmissable? Or is anyone really looking forward to something we haven't seen yet?" Let their answers be your guide to what to do next.
You'll have a marvelous time. You won't miss a thing.