No, the 320 isn't new, but it's a lot newer. It benefits from a wider fuselage (more passenger room, bigger overhead bins), fly by wire, better short field and high altitude performance, faster speed, lower fuel consumption.
The landing gear on the 737 is its Achilles heel - short to save weight and money, it can't be easily lengthened. Ever wonder why the engine nacelles on a 737 look squished and not round? They are. Because the gear was so short that turbofans, when they came out in the 70s wouldn't fit under the 737 wing wit ou squishing the intake.
Short landing gear, on the stretch version of the 737, severely limits the body angle for takeoff and landing, leading to ridiculous rotation and landing speeds, and severely degraded performance.
That's why 737s going to Hawaii take off with empty seats, the 12,000 foot runways at places like LAX don't allow the 737 to get going fast enough to carry their full weight into the air with the restricted angle of attack created by the geometry of that stubby gear. They just can't make enough lift.
So, in Mexico City, where an A 320 can take off at full gross weight, the 737 is weigh restricted and has to take off with less than a full passenger load, and only with enough gas to go short range. In Houston, where a 320 with the same number of passengers is landing at 125 knots, and can easily turn off the runway after landing, the longer 737s are landing at 165 knots, scorching their brakes, barely able to stop in 10,000 feet. That 40 knot increase in approach speed is a 70% increase in kinetic energy and stopping distance. That's a huge performance limitation.
United has had three airplanes slide off the runway/taxiway in the past two months.
Care to guess what aircraft type all three of them were?
Southwest has, famously, slid a few airplanes off the end of the runway...what airplane type do they fly?
Guys love the airplane if they never knew anything different. Take a kid that's never driven a car, and the Vega would be totally awesome...but compare it to something decades newer and the Vega (and the 737) don't compare well. Updates to the 737 are like adding Bluetooth and cool rims, it's still the same clunky, limited, economy car design underneath. Can't change the limitations of the chassis.