The cause is the nature of Tiny Core.
You really should read the Core book.http://tinycorelinux.net/corebook.pdf
If the connection has been configured via network tool, then a script with the name of the interface (e.g. eth0.sh) is stored in /opt and a respective entry is written to /opt/bootlocal.sh (so it's started at boot).
As always, refreshing the backup is needed in order to survive reboots.
The directory /opt is configured to be included in the backup by default.
If you have created the connection manually, directly in the console via commands, then there is nothing to be saved.
If you have created a script to start your connection, then you have to add this script to your backup manually (except for files in /opt) and create a respective entry, for example in /opt/bootlocal.sh , if you want it to be started at boot.
Services are managed differently on Core:
sudo /usr/local/etc/init.d/dnsmasq restart
For most services, you can also use the services tool.