study the examples in this dictionary link it will be the same in any dictionary. https://dictionary.cambridge.org/dictionary/english/strength the use of the word Might or Strength would depend upon what you are trying to convey. Might = power and force. (strength is therefore implied) used to describe armies and gods and mythical creatures or large powerful animals like a full grown male lion. Strength = the ability to endure and to continue to overcome. Example sentences. "the mighty army defeated the farmers rebellion of 2015" the strong army does not work in this sentence "he had the strength to carry on and complete the marathon" the might to carry on does not work in tis sentence. "she has the strength of character to succeed" the might of character does not work in this sentence. (remember strength of character it is a well used collocation) "lord who made has mighty make us mightier yet" lord who made us strong make us stronger yet does not work or sound right, it will not get the emotional patriotic response the song was written to achieve. "the old dam is weakening a new stronger dam needs to be built" a mightier dam does not work in this sentence "The Mongolians once had a mighty army" a strong army does not work in this sentence. "if you elect me as president/prime minister I will build a strong economy" a mighty economy does not work in this sentence. "if you elect me I will make this nation mighty once again" aimed at older voters "if you elect me this nation will be strong in the face of opposition and we will not hesitate to strike a mighty blow to those who seek to destroy our way of life" aimed at everybody

They basically mean the same thing. Might means impressive power and strength. So if he was pushing with all his might, he was pushing with all his strength and power. Although if he is a mighty man, he has more power than a strong man has.