Difference between $38 5W20 vs $18 10W30 oil change?

2002 Honda civic with 128,000 miles that has used 5W20 for 14 years with the dealer. Finally sick of overpaying.
The Honda dealer wants $38 for oil change for 5W20. The local shop offers 10W30 for $18.
What happens if I go with the 10W30 if anything?
it's not exactly a fresh ride, lol.

With modern engines using the correct oil is pretty important. They have much tighter clearances and things like v-tec valve timing that rely on the correct oil pressure to activate. As well as the viscosity (the 5w20 rating) the is also the API rating. This relates to the oil's actual lubricating properties. A 2000s vintage engine should be using at least an SL rated oil (newer SM or SN is OK too)

Now any old budget oil may be lower rated (SF etc). Fine in an 80s or older vehicle. But don't run that in a more modern engine.

Now you may be able to find cheaper oil that is OK. Wait until the auto parts store has some with the correct ratings on special, and grab a jug of it for ~$30. Exact brand doesn't matter, just check the viscosity and API rating is correct.

But don't risk your engine putting any cheap old sludge in there.

You should use the oil recommended by the manufacturer… Do you want to risk stressing / damaging the engine over $20?

If you are in a warm climate, the 10W-30 is fine.

In AZ we use the right oil. Synthetic blend 5w-20 oil changes cost about $20.00 at several places within 5 miles of me, coupons come in the mailers it's $25 everyday… Conventional 10W-30 is $18 everyday when you get there they'll normally tell you it's not right then upsell to proper. 10W-30 flows slower cold and is thicker hot… Harder for the oilpump to move and if you have VTEC the valve actuati… Heck there's no justification if you live on earth that has roads 5W-20 is fine summer or winter…