Azure Recovery Services and SQL 2014 Managed Backups not playing well together

It sounds as if what you are looking for differential backups. Those would contain everything added to the database following the last full backup.

I.e. you take a full backup on Sunday evening, with a differential backup every day after. On Monday evening, your differential backup would contain everything added since the backup. On Tuesday, it'd contain everything it contained on Monday, as well as everything changed since then.

If you'd do the same using Transaction Log Backups, your Monday evening backup would actually be identical to the differential backup described above. However, the Tuesday version of the Transaction Log backup would only contain the changes from the time of the Monday transaction Log backup.

When it comes to restoring, this would mean that in order to restore to a point in time, you would have to restore the latest Full Backup (Sunday), followed by every transaction log backup since, in sequence (Monday, Tuesday, etc.).

Using Differential backups, though, you would restore the latest Full Backup (Sunday), followed by the latest Differential Backup (Tuesday, if you're restoring on Wednesday).