Which words in the German language are written exactly identical, but take completely different meanings when used with different articles?
Sure! Here is a sample list of German words that are written identical (homographs) but take different meanings based on the article that is used:

1 - das Fliegen vs die Fliegen (nominalization of 'to fly' vs 'the flies')
2 - der Mann vs dem Mann ('the man' in the nominative case vs '(to) the man' in the dative case)
3 - das Band vs die Band ('the band or ribbon' vs 'the (musical) band' an English loan word)