"Regret" vs. "remorse"

The meanings overlap, but they are not the same.

An easy way to get the idea is that regret is for doing something stupid while remorse is for doing something immoral.

You lend your neighbour some money and he leaves town without paying it back - you may regret lending him that money, but probably wouldn't feel remorse.

You have an affair and get caught, your wife is heartbroken, then you may well feel remorse for what you did (though possibly not regret!)

Remorse is deeper and stronger than regret. One sense of regret is

A feeling of sadness, repentance, or disappointment over something that has happened or been done.

You may have associated regret with Italian "rimorso" because a deed was done. But regret is not limited to the above sense; it can also be used to talk about deeds that weren't done, ie also has sense

A feeling of sadness, repentance, or disappointment over something that hasn't happened or hasn't been done.

Regret thus seems to correspond with your "rimpianto" in that, in your example, you focus on the deleterious consequences of an action (or inaction).

As well, regret may be used of acts (of commission or omission) performed by others.

Remorse addresses the speaker's moral sense of personal guilt. It is directed inward, a gnawing feeling of distress over one's own responsibility for one's own grave error—in many cases a sense of having sinned; while regret is directed outward—an unmoralized feeling of distress over the unhappy consequences of an act which may have been performed (or omitted) by anyone.

So remorse can be your rimorso if, in your example, you are using the same sense of "doing something wrong."

I would like here to expand a little the research that maybe someone could find interesting. As correctly explained above, "rimorso" in italian is the pain for something done, similar to the english "remorse". But, differently, it doesn't mean repentance, repentance is a desire to repair (which is a step further). "Rimpianto" is the pain for something lost or not done, and derives from the word "pianto" which means "cry". The link between the english and the italian for the word regret is the verb "regrettare":

regrettare v. tr. [dal fr. regretter «rimpiangere»], ant. e raro. – Rimpiangere: alcune anime sensibili e filosofiche regrettarono persino l’antico stato selvaggio (Beccaria).

Which comes from the french "regretter", to cry for something that is lost.