I'd say definitely update the DL since having "Real ID" you won't need to carry around your passport for domestic flights. Your license term will now be like 5+ years, instead of 2 years. Also, your SSC number will stay the same unless you had identity theft. If for some reason, you had a change in SSC, any license or credit cards, etc. you'll need to update those. That's a headache.

Anyways, congrats on getting your GC!