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!
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8/16 Application Sent to Phoenix, AZ
8/17 Application Delivered
8/21 i797C's sent out for i765/i821-d
8/23 Biometrics letter received, scheduled for 09/13
9/13 Biometrics completed 10/5 -APPROVED