EDIT: When I check the status using "tvservice -s", then a difference is shown. But I cannot see any difference on the TV visually.
There shouldn't be any visible difference if TV respects the signalling.
The setting will change both the (limited/full) range of the pixels output, and the signalling of limited/full range to the TV.
So if the TV understands the signalling (it should) then it should display very similar pictures in both cases.
The setting is only useful when the TV fails to respect the signalling, and e.g. if interpreting the pixels as full range when they are actually limited.
Forcing the pixel encoding will solve that, but it is rarely required (only when TV is causing the problem).
So, I don't think this is the issue.
The TV may have different presets for colour/contrast/brightness from different sources. You may want to experiment in the TV menu.