Regret is sad for themselves, remorse is sad for you — is a slight oversimplification — yet, covers it for the most part.
All good answers so far, what OldWOunds said however is pretty much it in a nutshell, IMO anyway.
Statements of regret, when you break them down, are about the other person, and more specifically, about how what they did affects THEIR feelings. There is nothing wrong with regret, it's a precursor to remorse, but it is still inward-facing. "I feel like such a piece of shit for cheating on you, I wish I had never done it, I'm so sorry for cheating" is regret. It addresses their feelings but doesn't mention or consider you at all.
Remorse is taking regret and combining it with empathy in order to understand how the things you regret affect other people. "You deserved better than the way I treated you. You put your faith in me and I disregarded your feelings. You have every right and reason to be mad at me forever".
A mix of the two is common and in most cases, the best approach, as it can be helpful for the victim to know what their abuser is thinking and feeling, especially as it relates to their own emotions. "I'm such a POS for cheating on you, you deserved better than that."
Your gut is probably your best guide however. If you have a conversation with your spouse and you end up feeling unseen and unheard, then your WS is probably still stuck in shame and regret. If you feel as though your feelings were recognized or validated, then you're probably experiencing remorse from the other person.
Me: WS
BS: ISurvivedSoFar
D-Day Nov '16
Status: Reconciling
"I am floored by the amount of grace and love she has shown me in choosing to stay and fight for our marriage. I took everything from her, and yet she chose to forgive me."