The difference in Healthcare systems certainly plays a part but I think the most critical thing here is that, broadly speaking, Germans do as they're fucking told.
Culturally speaking the way various laws work in germany means that as a society they all are more aware of the individual's responsibility to society as a whole and they are much better at following guidance/advice and laws than people here ever have been.
The biggest factor in their reduced stats will be that the nation took it seriously from the start and social distancing, self isolation and other measures were implemented quickly and pretty much iniversally. none of this 'fuck it let's go down the pub' bollocks.
Hmm, I read the German press (am a Gm native), and they have been printing exactly the same articles about citizens being irresponsible in terms of social distancing, and complaining about a lack of detail and clarity in the instructions they're receiving. They have now got the message and are printing pics of empty streets, just like the British press, so I'm not sure there's a huge difference in that respect.
There are also big regional differences in Germany as the Länder have much greater legislative autonomy than is the case in the UK, so some regions have acted much more firmly than others. There's also been a big debate and philosophical angsting (more than here) about balance between safeguarding democratic rights and constraining people's freedoms of choice and movement.
I suspect more proactive testing and better healthcare infrastructure are likely to be bigger factors than some vague national personality characteristics - Germans definitely do have different attitudes to some rules than Brits or Italians, but it's not Singapore or China, where the govt issues orders and the population just get on and do what they're told.