No difference. None at all, and anyone who says otherwise is full of schitt.
Here's the bottom line: You're either together with someone, or you're single. Together means only one thing, and single means only one thing. Unless you're legally married to someone, there is no in-between.
The status of "taking a break" was invented by neurotic young girls who wanted have their cheque and cash it too, and it's total bull shift. It's a direct and deliberate violation of Guy Code Rule #112, which clearly states "Never listen to a woman complain longer than she's willing to engage in sexual activity with you, use a stopwatch and hang up the phone if necessary.". That's a rule for a reason.
When a girl says the relationship is "on a break", she means you have to listen to her complain (and open jars for her, and carry stuff for her, etc. etc.) but she doesn't have to unbutton or unzip anything for you.
Completely unacceptable. If your boss fired you and then asked you to come in to work for free until a replacement could be found, would you agree? If someone asked to buy your car but couldn't pay you any money until next month, would you sign it over that day? Exactly. But that's what a girl expects from you when she says "we're on a break". It means she's not getting naked with you, but you're not allowed to get naked with anyone else the whole time.
There's a word for being together but never having sex, and that word is marriage.