Does the difficulty of German language exams vary from city to city?

For the standardised exams, no. For the university-run DSH, partly.

• TestDaF is developed and graded centrally by the TestDaF-Institut; the paper is the same on each exam date regardless of the test centre, in Germany or abroad.
• Goethe-Zertifikat and telc exams are standardised too: the same format and assessment criteria everywhere, with central or centrally trained examiners.
• DSH (Deutsche Sprachprüfung für den Hochschulzugang) is different: each university sets and runs its own DSH within a common framework agreed by the universities (HRK) and the education ministers (KMK). Format, topics and strictness therefore differ between universities, and candidates do report that some DSH exams feel harder than others. A passed DSH is nevertheless recognised nationwide.

What really changes from place to place is access: waiting times for TestDaF dates, how often a university offers the DSH, and whether a DSH is open to external candidates.

If you want the most predictable route, take TestDaF or a Goethe C1/C2 exam; if you take the DSH, practise with the sample exams of the specific university.