The carriers have blurred the distinction. AT&T has been the worst offender, labeling their LTE-Advanced service as "5G E".

5G is the fifth generation, which will provide the everyday capability of gigabit service and low latency. As has been mentioned, this will be mostly useful to the internet of things (allowing another order of magnitude-worth of stuff to connect).

Both T-Mo and Verizon have been making big strides in deploying 5G. T-mo does it using low-band frequencies, principally their 600 MHz spectrum, which has limited capacity but good reach. 5G on T-mo will mostly feel like fast LTE. Verizon has done it using millimeter-wave frequencies (around 60 GHz I think) which has only a few hundred feet range at best but can support much faster speeds.