Deleting a backlink is simply deleting the backlink from the Monitor Backlinks system. The backlink still exists on the internet.
Disavowing a backlink is telling Google that you don't want to be associated with the backlink, so that you don't potentially get penalized for it. Again, the backlink still exists on the internet—Google cannot change that.
If you disavow a backlink in Monitor Backlinks, you still need to submit a disavow report to Google to finish the disavowal process. This article has more details on that.
For the most part, you should not really delete any backlinks from Monitor Backlinks. The only reason might be if you want to reduce the number of backlinks tracked so that you don't need to upgrade your plan. (This will not really work long term though, because those links will get re-added as our system discovers new backlinks.)
And in fact, you should not delete backlinks even if you have disavowed them. It's best to keep track of them so that the next time you want to disavow more links, you can simply update your disavow report.
If you have previously disavowed links, you can export the old disavow report from your Google Disavow account and then upload it to Monitor Backlinks here.