If the govt takes the responsibility out of your hands there certainly is a practical difference. I dont call having the govt forcing you to do something taking responsibility. If you parents forced you to eat your peas, is that you taking responsibility for eating healthy?
This is in many ways just semantics, but now you are saying 'taking responsibility' as opposed to talking about 'personal responsibility'. They are not the same thing as one is an affirmative choice and the other is simply a fact of being. The government is most certainly not taking responsibility out of people's hands, it is in fact putting responsibility in them through the individual mandate. 'Personal responsibility' is not dependent on whether the person sought it out or someone else forced it on them.
In your example there is of course a moral difference between choosing to take on a responsibility as opposed to only doing so under duress and in that I agree with you. That's not really what we're talking about here though. If you sue someone for running you over with their car and win a settlement for your medical bills, you have forced them to take responsibility for the damage they did. Sure it's not as good from a character perspective as if they had volunteered that money themselves, but you could hardly say that they weren't held personally responsible for their actions.