When one hose to heater core is hotter and the other is much cooler what would this indicate 2004 impala?
Maybe that heater is air bound and needs to be bled. Remove "cooler" heater hose and let water run through heater until water runs hot--be careful HOT Water!!!
More likely is that the heater core is clogged. Mark the hose that is cool - this hose is the return from the heater core to the engine. Let the engine cool, carefully release any pressure in the system by slowly removing the radiator cap. Remove the two heater hoses at the engine block. You can adapt some garden hose fittings into the hoses. Connect your water supply to the heater hose you marked, connect a length of hose to the other heater hose and place the end into a 5 gallon bucket. This will allow you to see anything that is flushed out of the heater core. Slowly open the valve on your water supply. You are now reverse flushing the heater core, and probably have seen a bunch of crud enter your bucket as soon as you started the water flow. Flush until no more crud is visible entering the bucket, remove the garden hose hookups, reinstall the heater hoses at the engine, fill your radiator to the top, reinstall the radiator cap, start the engine and let it warm up to operating temperature. Check for heat.