[mesmark]

I was running some examples through my head (a completely trustworthy and scientific method :P) and what I find is that I tend to use "compare to" when I'm showing contrast and "compare with" when I want to express similarities.

"This is nothing compared to the storm of '97."

"Compared with the storm of '97, we see that high pressure systems preceded both disasters."

So in your examples I would probably use "to" in both examples:
1. Teaching adults is more enriching compared to teaching kids.
- the sentence implies they are very different - contrast

2. I am trying to compare the movie The Day After Tomorrow to 2012.
-this example is a little less clear for me. I need the context of surrounding sentences but I'd most likely use "to"

Anyway, I still stick with they are rather interchangeable. Those are just some thought I had on the matter.

[Denis DNT]

Thanks Susan and Mesmark. It had never crossed my mind to think about the two until recently a student asked me in class. My colleague insisted "to" was more in use than "with".