Not sure about more “energy”, as your wifi router likely consumes more electricity than your phone charger. But yeah, battery life can be better because the wifi signal is shorter range than the cell tower signal, and thus your phone will require less antenna power to communicate.
The biggest effect is on your data usage though. I have a regular job (wifi on site) and a wifi router at home, and with these two I barely use 500 MB of data, so I can go with the cheapest data plan and be ok.
You should be able to see your data usage on your phone, so monitor it with LTE all month and then with wifi (as available) all month, and see if you could save a ton by switching to a plan with a lower data cap.
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