to remember: to call something back to mind, or to hold someone in mind and act on their behalf
Most of the time this verb is simple recollection: a fact, a word, or a promise that was known before suddenly comes back to mind. The disciples remember what Jesus said only after the resurrection makes sense of it, and Peter remembers the warning about the rooster the moment it crows.
But the word also reaches past the mind into action. To be mindful of a prisoner is to care about him and do something, not merely to think of him. And when God is the one who remembers, the word carries its full biblical weight: God remembering his covenant or his mercy means he is stepping in to keep his promise, not that anything had slipped his memory.
The same active force runs the other way with sin. For God to remember sins is to bring them to account, as when judgment finally falls on Babylon. So the promise that he will remember sins no more is not divine forgetfulness but full forgiveness: he refuses to hold them against his people ever again.
These are shades of one idea rather than separate words. Whether the subject recalls a fact or God acts on a promise, the core is the same: something or someone is brought back into active attention.