If you're talking to someone on the phone, they mean the same thing. I would say you're fine using them interchangeably. Here are some examples: I can't understand you. Your voice is cutting out. I can't understand you. You're breaking up. Can you repeat that? Your voice was cutting out. Can you repeat that? Your voice cut out for a moment. Can you repeat that? You were breaking up. Did you hear what I said? Or was I breaking up? Did you hear what I said? Or was my voice cutting out? I think we have a bad connection, so I might start breaking up. I think we have a bad connection, so my voice might start cutting out. There isn't a specific rule but I find that when you're using the word "voice" in the sentence you tend to use the "cut out" phrase, and when you're not specifically saying "voice" (but it is implied), you use "break up". But again, this isn't a specific rule, it's just what seems more natural to me as an English native speaker from Texas.

"Cut out" means that the connection is lost. We would be more likely to say "cut off" for a phone call "Breaking up" means that there only part of the digital signal is being received so rather than hearing, say, "Good morning" you might hear "g---d -or--ng" Source: https://forum.wordreference.com/threads/voice-cut-out-voice-break-up.3355592/

To me, they are mostly interchangeable. The slight difference would be that 'cut out' refers to a period of silence, whereas 'breaking up' means that the signal is coming and going. You hear bits of sound, then bits of silence. 'breaking up' is more intermittent.