Why isn't the Iptables persistent service saving my changes?

iptables-persistent does not work that way. Restarting the iptables-persistent "service" does not capture the current state of the iptables and save it; all it does is reinstate the iptables rules that were saved when the package was last configured.

To configure iptables-persistent, you need to tell it about your current iptables ruleset.

One way to accomplish that is as follows:

iptables-save >/etc/iptables/rules.v4
ip6tables-save >/etc/iptables/rules.v6

Or, equivalently, the iptables-persistent package also provides the following:

dpkg-reconfigure iptables-persistent

(You will need to answer yes to the questions about whether to save the rules.)

After that, the next time iptables-persistent is started/restarted, the iptables rulesets you expect will be loaded.

Very simple way to save the current iptables rules is to use the command:

sudo service netfilter-persistent save

Using the above, which works at least in Ubuntu after installing the netfilter-persistent (and iptables-persistent) package, there is no need to run manually the iptables commands or to reconfigure the package (as suggested by even the accepted Answer above).

Install iptables-persistent:

sudo apt install iptables-persistent

save rules after making desired changes:

sudo netfilter-persistent save