How does using coffee filter make the coffee different? Somewhere I read that coffee contains carcinogens unless it's filtered, and I think I've noticed that such filtered coffee seems to be weaker. I let it stay for about 5-10min, and it still tastes weaker than coffee made without a coffee filter (paper one of course).

So my question is, how is a filtered coffee different from unfiltered coffee, assuming we're using a paper filter? Mainly I'm looking for a scientific explanation.
There's two points to address.


"Weaker" coffee with paper filters as compared to brewing right in the cup. As far as I understand (although I can't find citation), paper filter does not capture caffeine or aromatic compounds very much. The feeling that filtered coffee is thinner may in your case come from less microgrounds floating in the cup (i.e., from mechanical, rather than chemical, sensation).
I'm not sure what you meant by carcinogenic properties, but you may want to read about cafestol and kahweol, two diterpines that are supposed to have anti-carcinogenic effect. They are also said to raise cholesterol levels (this study). They are removed by paper filters, and if you want to go into details of its levels in various preparations, see this article. It reports that cafestol is lower in darker roasts, and among the non-filtering methods is highest in swedish boiled coffee and lowest in moka pot ("stovetop espresso").