The Corded Mobile Connector is a variant on Tesla's original UMC, with a fixed 14-50 plug attached.
The original UMC (now called Mobile Connector) supported up to 50A outlets (40A of charging).
The Gen 2 Mobile Connector will connect to a 14-50 outlet, but is limited to only 32A of charging; resulting in some cost savings by using a lighter, lower rated cable.
We're using the 14-50 Corded Mobile Connector with our S P85, and had planned to use it on our new Model 3, until we decided to get a Model X instead.
When planning for your charging, consider that you typically want to keep your car between 10-90% of charge. That means the max charge needed overnight is 80% (with most driving using far less than that in a single day).
Assuming that the Model 3 Long Range battery pack is 75 KWhr of capacity, that means the max you would likely charge overnight would be 60 KWhr.
If you use a Gen 2 Mobile Connector at 32A on a 14-50 outlet, you'll get about 7 KWhr of charge per hour, which would take about about 8.5 hours to go from 10% to 90%.
If you used the Corded Mobile Connector, you would be able to get 40A of charging, about 8.5 KWhr of charge per hour, and reduce your 80% charging time down to about 7 hours.
Since most daily driving will use far less than 80% of the charge of the battery pack, even with a 32A Gen 2 connector, you'll be able to fully recharge overnight - which is probably why Tesla decided to save some $$$ on the Gen 2 connectors by only supporting 32A.
With our 2 Model S cars, we carry the UMC in the car, for an emergency, like carrying a spare tire. Other than on a road trip, when we're staying at someone's house or at a location with a 14-50 outlet, we've never used the UMC - but we still carry it around, and will do the same with our Model X.
Getting an HPWC, an extra Gen 2 connector or the 14-50 Corded Mobile Connector and mounting it on the wall (using the cable organizer for the mobile connectors, no longer needed for the new HPWCs), makes it easy to charge at home.
And based on our experience with our Model S's over 5 years, if you wanted to save $$$, you could probably just use the Gen 2 connector that comes with the car for home charging - and then take it with you when you go on a road trip.
