Banshee has had way more memory wipes than Cayde-6 and as the grimoire attests [url=http://www.ishtar-collective.net/cards/gunsmith?highlight=banshee-44]his memory has suffered.[/url]

Cayde states that memories do survive the reboots. It's how he remembers some parts of being human and how he became an Exo. He was offered a job with Bray, one that seems to have required him becoming an Exo.

Imagine your brain works like a computer. Organized into neat files and documents. You can tell if you've been wiped based on things like gaps where there should be things, or time stamps being all wrong. Of course it may take some time to notice, but you'd notice if there was an odd blankness to your memory. As far as remember those memories, perhaps he just dragged them out of the recycle bin? lol.