How do those terms differ from each other?
downtime
North American A time of reduced activity or inactivity:
everyone needs downtime to unwind ODO
spare time
Noun
- time available for hobbies and other activities that you enjoy: free time
time off – a time period when you are not required to work; he requested time off to attend his grandmother's funeral
- time that is free from duties or responsibilities: free time
leisure, leisure time – time available for ease and relaxation; his job left him little leisure WordNet by Fatlex
Please, compare:
Everyone needs downtime to unwind.
Everyone needs time off to unwind.
Everyone needs free time to unwind.
Everyone needs spare time to unwind.
-and-
Annette and her husband use their free time to attend Chicago Bears games, play with their puppy, and spend time with their new granddaughter Annabelle.
Annette and her husband use their spare time to attend Chicago Bears games, […]
Annette and her husband use their time off to attend Chicago Bears games, […]
Annette and her husband use their downtime to attend Chicago Bears games, […]
Best Answer
"Free time" and "spare time" are mostly synonymous and imply that your existing or habitual schedule has leftover time that you can allocate as you see fit. The dictionary notes about hobbies or "free from responsibilities" is referring to this.
"Downtime" and "time off" both imply some form of disruption in your existing or habitual schedule. The key difference between these and "free time" is that "downtime" is a change and "free time" is part of the normal schedule.
The difference between "downtime" and "time off" is mostly in connotation but for the scope of this question they are synonymous. If you are curious about the nuances between those two in particular, feel free to ask a separate question.
To explicitly address your examples:
These mean, "Everyone needs to take a break from the normal schedule to unwind."
These mean, "Everyone should allocate time to unwind in their schedule."