Does the manga continue the story from the anime?
The short answer is no, and the anime does in fact end (so why would there be a second season?).

The manga and anime are similar, but they diverge and quite significantly at some point so that you might consider them alternate tellings of the same story.

As Aki pointed out, there are a number of differences between the manga and the anime version.

It's been a while since I've seen and read them, but I believe the overall story is more or less the same except for the ending.