Remorse is feeling sorry for what you did. It may be that you regret what you did or not. It does not include a need to confess but does indicate a pang of conscience.
Repentance includes regret for what you did and a desire to reverse the situation if it were possible. There is also a need to make things right and a determination not to repeat the crime. With repentance, you must confess your sin in order to move forward and make amends for what you did.
Both include intense feelings of guilt.
Lady Macbeth is indeed haunted by her actions and definitely remorseful. In this sleepwalking scene, she does basically confess her sin - "Who would have thought the old man to have had so much blood in him?" - but her rantings only serve to confuse the doctor and attendant.
The doctor knows that
Do breed unnatural troubles; infected minds
To their deaf pillows will discharge their secrets
but he cannot understand or contemplate the extent of her collaboration. It is too terrible to "think, but dare not speak."
It is certainly true that it is her recognition of the extent of what she and Macbeth have done that drives her to suicide
the stain of sin having eaten away at her mind
but as she is no longer aware of her surroundings, she, despite confessing, cannot be said to have repented. She seems more to need to confess in an attempt to clear her conscience (which clearly fails) rather than out of any real regret.
Lady Macbeth does, even in her time of madness, still feel protective towards Macbeth, attempting to comfort him (even though he is not there )
Wash your hands, put on your nightgown...Banquo's buried
and does make the reader or audience feel some compassion towards her but again is not indicative of repentance. It is perhaps her only nurturing and womanly feature but she has concern for her husband ,not sorrow for what happened. She worries that he will be caught but her torment is not caused by a wish that they had never conspired but that it causes so much guilt to surface - something she did not anticipate, having supposedly mentally prepared herself well in advance. She would even feel nothing at killing
a baby at her breast to honor a vow.
She consistently steels Macbeth to proceed with his ambitious plan to be king - with or without the witches help. Knowing that as long as the are careful "a little water clears us of this deed."Nobody will even know what they have done. Perhaps losing Macbeth's trust is part of the reason for her descent into madness as she is no longer needed by him. Definitely not a repentant woman!