The difference between 比～更 and 比～多了
Simply, 


  比～更: even (more)；
  
  比～多了：much (more)；


他打篮球比我更好。： he plays basket ball even better than me. 

他打篮球比我厉害多了。: he plays basket ball much better than me.

坐高铁比坐飞机更方便。: Taking high-speed railway is even more convenient than taking a flight. 

坐高铁比坐飞机方便很多。: Taking high-speed railway is much more convenient than taking a flight. 

Updated according to the updates from OP:

Compared these three sentences: 


  
  他打篮球比我好。
  他打篮球比我更好。 
  他打篮球比我厉害多了。
  


2 and 3 are stronger than 1. 2 and 3 are hard to compare, because they are used for different purposes.