What's the difference between capital punishment and the death penalty?
"Death penalty" applies to a prisoner who has been sentenced to die, but has not yet been executed; "capital punishment" refers to his actual execution.
When the prisoner is actually executed, we refer to this as capital punishment. When execution is pending, looming, or just being threatened to coerce a plea, it is referred to as the death penalty. Prosecutors often offer to take the death penalty off the table in exchange for a confession to avoid or shorten a lengthy and expensive trial.
A lot of murder prosecutions never go to trial or end in some sort of plea bargain before the jury renders its verdict. Many states don't actually execute anybody, even the prisoners who were at some point sentenced to be executed. A death sentence triggers a long appeals process, which rarely results in acquittal but frequently results in a sentence being changed to life imprisonment or some lesser sentence.
Charles Manson received the death penalty 45 years ago and still sits in prison, alive (Edit: He finally died in 2017). Despite being one of the most notorious murderers in American history, Californians' attitudes toward capital punishment changed while he was on Death Row, for reasons that didn't really apply to him directly. California eventually brought the death penalty back, but Manson doesn't face it because his sentence was commuted to life imprisonment in 1972.
If you were to use the two terms interchangeably, no one would correct you.