What differences are there between the implications for immigration status between marriage to different nationals?

Yes, that's correct. It is the case in the UK and a few EU countries. In particular, the UK family visa has English-language knowledge and income requirements and significant fees (£1,000-£1,500 per person), which are all illegal under EU freedom of movement rules.

All this is the basis for the Surinder Singh route but it is presently unclear how much will subsist after the end of the Brexit transition period (probably not much). Cases like this have been cited in some circles as a reason for Brexit.