Deleting a backlink is just deleting the backlink in the Monitor Backlinks system. The backlink still exists on the internet and we could not change that.
Disavowing is telling Google that you don't want to be associated with a backlink. So you don't potentially get penalized by it in the eyes of Google. Again, the backlink still exists on the internet and Google could not change that. If you disavow something in Monitor Backlinks, you still need to take submit a disavow report to Google to finish the disavowal process. Here are more details on that: http://intercom.help/monitor-backlinks/disavowing-links/how-do-i-disavow-backlinks-using-monitor-backlinks
For the most part, you should not really delete any backlinks within Monitor Backlinks . The only reason would really be if you want to reduce the number of backlinks tracked so that you you don't need to upgrade your plan to continue using the features. (This will not really work long term though, because those links will get readded as our system discovers new backlinks.)
And in fact, you should not delete backlinks, even if you've disavowed them. It's best to keep track of them, so that the next time you want to disavow some new links, simply update the old disavow report. If you disavowed previously, you can go to your Google Disavow account, and export the old disavow report and then upload it on Monitor Backlinks here: https://monitorbacklinks.com/app/disavow