console.error() writes to stderr, whereas console.log() writes to stdout as described in the doc.
In a default run of nodejs, both stdout and stderr go to the console, but obviously, they could be directed different places and could be used differently. For example, when using tools such as forever, the two streams are logged to separate log files so they can be examined separately.
The presumption is that console.error() may contain more serious information that might want to be looked at separately, but that is really up to how you use console.log() vs. console.error() in your own program.