The first reason is that the bug interacts with other software in unexpected ways. We know that interacting with the shell is dangerous, but we write code that does it anyway. An enormous percentage of software interacts with the shell in some fashion. Thus, we'll never be able to catalogue all the software out there that is vulnerable to the bash bug. This is similar to the OpenSSL bug: OpenSSL is included in a bajillion software packages, so we were never able to fully quantify exactly how much software is vulnerable.
The second reason is that while the known systems (like your web-server) are patched, unknown systems remain unpatched. We see that with the Heartbleed bug: six months later, hundreds of thousands of systems remain vulnerable. These systems are rarely things like webservers, but are more often things like Internet-enabled cameras.
Internet-of-things devices like video cameras are especially vulnerable because a lot of their software is built from web-enabled bash scripts. Thus, not only are they less likely to be patched, they are more likely to expose the vulnerability to the outside world.
Unlike Heartbleed, which only affected a specific version of OpenSSL, this bash bug has been around for a long, long time. That means there are lots of old devices on the network vulnerable to this bug. The number of systems needing to be patched, but which won't be, is much larger than Heartbleed.
There's little need to rush and fix this bug. Your primary servers are probably not vulnerable to this bug. However, everything else probably is. Scan your network for things like Telnet, FTP, and old versions of Apache (masscan is extremely useful for this). Anything that responds is probably an old device needing a bash patch. And, since most of them can't be patched, you are likely screwed.
Update: I think people are calling this the "shellshock" bug. Still looking for official logo.
Update: Note that the thing with the Heartbleed bug wasn't that that the Internet was going to collapse, but that it's in so many places that we really can't eradicate it all. Thus, saying "as bad as Heartbleed" doesn't mean your website is going to get hacked tomorrow, but that a year from now we'll be reading about how hackers got in using the vulnerability to something interesting.
Exploit details: The way this bug is exploited is anything that that first sticks some Internet parameter in an environmental variable, and then executes a bash script. Thus, simply calling bash isn't the problem. Thus, some things (like PHP apparently) aren't necessarily vulnerable, but other things (like CGI shell scripts) are vulnerable as all get out. For example, a lot of wireless routers shell out to "ping" and "traceroute" -- these are all likely vulnerable.