@frank-s Think of it like this...
You take the original backup with snapshot on October 1st.
You then have a differential backup on October 8th. That differential only writes the changes that make the state of the server on October 8th different (hence the term differential) from the state of the server as it was on the full snapshot on October 1st.
Now you delete the snapshot of the full system state taken on October 1st, leaving only the October 8th differential.
It's now October 15th and you want to restore something from the October 8th differential and you figure you'll just make a new full snapshot from October 15th and then try to apply the October 8th differential. However, this won't work because the differential from October 8th will be missing key reference points it will be looking for in order to overlay its changes onto the snapshot. What else on that server changed from October 1st to October 15th that isn't contained in the differential? What changes occurred from October 8th to October 15th that are newer than what is on the differential? You could, potentially, brick your entire system if the differential started overwriting changes it sees from its system state that are actually newer than the data it contains.
Differentials are entirely dependent on the full snapshot on which they are based. Any new snapshot taken after their creation will be totally foreign to the differential... those differentials will be looking for very specific system states that existed at the time the original snapshot was created, and those will be completely different from the system state as it will appear on a new snapshot created after the differential.