Hi Constantin,
Did I get it right that you are working with web application and going to test it using emulation of mobile web browsers? (This seems to be the only thing that you will be able to do with Web module.)
Well, actually, TestComplete and Selenium are pretty different in terms of tests architecture but you asked not to touch this. :)
From the technical point of view, I would not expect any significant problems with TestComplete to work with web application. (Though it definitely worth some initial evaluation as a lot of things may depend on the tested web application internal design and implementation.)
If you decide to try TestComplete, I would recommend to try your best to not search web elements by XPath (like in Selenium) but use .FindChild/.FindAllChildren/other .FindXXX and .WaitXXX methods provided by TestComplete.
The good thing that I like in TestComplete is that you always can correlate the web element referenced in test code with the one displayed on the screen (via the .FullName property). This is something that is hardly achievable in Selenium.
TestComplete does not have something similair to Selenium's grid. Thus you will need to either consider Network Suite functionality of TestComplete or manually dispatch where your tests should run. One more option is to consider CrossBrowserTesting integration provided by TestComplete. But it requires separate license.
When running scheduled or CI-dispathed tests, like with Selenium you will need active intaractive user session on the test computer. Headless web testing is not supported by TestComplete. (For obvious reason, as TestComplete mimics actions of the human user but do not use jscript code on the page to trigger this or that action.)