Is it true that different variants of the same model aircraft don't require pilot retraining?
It appears to depend on how much changed. You do have to have some training between models. The selling point was that an airline with 737 pilots could fly the MAX variant with minimal retraining


  "The airplane is configured to be very common with the [737 NG]," Wilson, who is now retired, said, "and so a pilot can walk into here and will find everything he can just like he can in the [737 NG]."
  
  "FAA-approved this for two-and-a-half hours of computer-based training for the transition between the two aircraft," Wilson continued. "All the overhead panel switches are the same. The only minor difference is, because of the change in the display, is to move some of the center console items here on the forward console." 


And from the Wall Street Journal


  To make the plane as inexpensive as possible for airlines to adopt, Boeing was intent on persuading regulators that pilots of earlier 737s should be allowed to start flying the MAX without simulator training. That training would have been required if there were substantial safety differences between the models, boosting the plane’s cost to airlines since training cuts into time flying paying passengers. 
  “Our marching orders are no training impact on this airplane. Period,” Richard Reed, a former Federal Aviation Administration engineer, recalled a senior Boeing official telling him during a meeting in the early years of the MAX’s development.
  
  The company had promised its biggest customer for the MAX, Southwest Airlines Co. , that it would pay it $1 million per plane ordered if pilots needed to do additional simulator training, according to Rick Ludtke, a Boeing engineer who worked on the jet’s cockpit systems, and another person who had been involved in the airplane’s development.


And later


  Boeing officials were focused on making the MAX fly as similarly as possible to earlier 737s. The fewer differences, the less likely the FAA would require pilots to undergo retraining.
  
  Mr. Ludtke, the former Boeing cockpit engineer, said company managers put pressure on engineers to avoid tweaks in designs that could result in pilots needing to learn new maneuvers in a simulator.


Wasn't there a difference?

The issue was that the MAX had a system (MCAS) that could push the nose down to make it behave more like previous 737s. Having read several articles on it, the root problem was


The initial specs said it was a minor change with the same course of action to correct as a runaway stabilizer (i.e. it was something 737 pilots already knew)


  The company reasoned that pilots had trained for years to deal with a problem known as a runaway stabilizer that also can force the nose of the plane to dip. The correct response to an MCAS misfire was identical. Pilots didn’t need to know why it was happening.

Boeing made MCAS 4x more powerful without really notifying anyone. It could also operate at slower speeds.
Boeing apparently wrote its manuals based on the original MCAS spec. Assuming it was still a minor system, it was buried or omitted


  Southwest Airlines, the jet’s first and biggest customer, followed Boeing’s lead and deleted MCAS from the manuals and emergency procedures it devised for its pilots. Other carriers didn’t mention it, either.

As predicted, the previous crew of the aircraft that would be Lion Air 610 had issues with MCAS, but correctly diagnosed a runaway stabilizer and shut off MCAS in the process


  The puzzled cockpit crew checked a quick reference handbook, running through other emergency steps before successfully regaining control of the plane by executing the checklist for a runaway stabilizer. That turned off MCAS and the crew flew manually for the rest of the trip.



Lion Air 610 would not be so lucky. They only figured it out moments before the plane slammed into the ground. Apparently it wasn't as obvious to pilots as Boeing thought.