Historically, ideas of "moral responsibility" are strongly linked with retributive punishment.
If you did something wrong, and you are morally responsible for it, then we are morally permitted (or even obliged) to make you suffer for it (where "suffer" can mean potentially anything from mild social opprobrium up to torture and execution). If you did something wrong, but (for whatever reason) lack moral responsibility for it, then it is morally wrong for us to make you suffer for it. If your moral responsibility is impaired but not completely absent, then it is wrong for us to make you suffer to the normal degree, but it might be justifiable for us to do so to a more limited degree.
But, many people today reject the idea of retributive punishment. And if you do so, it isn't clear how important the concept of "moral responsibility" still is. It is a logically coherent position to accept the evaluative/axiological aspects of morality (the labelling of states of affairs as "good" or "evil" or "neutral", their ranking as greater or lesser goods/evils), and its prescriptive aspects (you ought to do this, you ought not do that, you may do this but you aren't obliged to), while rejecting the concept of "moral responsibility" as misconceived, useless, harmful and/or erroneous.
If you did something wrong, and you are morally responsible for it, then we are morally permitted (or even obliged) to make you suffer for it (where "suffer" can mean potentially anything from mild social opprobrium up to torture and execution). If you did something wrong, but (for whatever reason) lack moral responsibility for it, then it is morally wrong for us to make you suffer for it. If your moral responsibility is impaired but not completely absent, then it is wrong for us to make you suffer to the normal degree, but it might be justifiable for us to do so to a more limited degree.
But, many people today reject the idea of retributive punishment. And if you do so, it isn't clear how important the concept of "moral responsibility" still is. It is a logically coherent position to accept the evaluative/axiological aspects of morality (the labelling of states of affairs as "good" or "evil" or "neutral", their ranking as greater or lesser goods/evils), and its prescriptive aspects (you ought to do this, you ought not do that, you may do this but you aren't obliged to), while rejecting the concept of "moral responsibility" as misconceived, useless, harmful and/or erroneous.