What's is the difference between these terms?

A death penalty is the legal provision for sentencing someone to death.
A death sentence is what is imposed on a person who is convicted of a crime that carries the death penalty
Death row is a special segregated area of a prison, where prisoners who are awaiting their death are kept. They do not stay in the general prison population once the death sentence has been passed.
Capital punishment means being put to death.

Death penalty & capital punishment are each the legal provision that a murderer will be put to death, although some may use each of these as a way to state the belief in that happening. Someone may say, "I believe (or do not believe) in capital punishment" or "I believe (or do not believe) in the death penalty." I could be wrong, but I think they're the same.

Death sentence is when the jury & judge declare that the person will be put to death, having met the requirement without mitigating circumstances, such as self-defense or temporary insanity or in some cases, that it was unplanned, or all sorts of legal loopholes.

Death row is the section of the prison where a person waits to be put to death or for a mistrial to be declared or to offer a legal challenge to the matter.

Generally, it's all of these mistrials & challenges to the process that often cost the state more than originally declaring a lifetime sentence without possibility of parole. In California's case, billions of dollars could be saved by repealing the death penalty & simply issuing lifetime sentences, but few voters are informed about that & continue to call for this "get tough" policy that they think saves the state money but doesn't.

Death sentence refers to the outcome of a particular trial. Death penalty is similar but it can also be used more abstractly. "The defendant received the death sentence." "I don't believe in the death penalty." Capital punishment is even more abstract. "Capital punishment was abolished in Slobbovia in 1969." They all can sometimes be used interchangeably."

"Death row" is the section of a prison where prisoners awaiting the death penalty are kept.

Here is the list as I understand it:
Death penalty - being executed
Death sentence - having a judge declare that you are to be executed
Death row - the place people who have a death sentence are kept
Capital punishment - any physical punishment (although I am probably confusing this with 'corporal punishment')

Keep in mind that people also use synonyms in writhing and other presentations so that they avoid boring their readers/listeners by using the same word over and over.