As far as I'm sure, the only connection is that you gain EVs is the same time gain experience through fainting an enemy Pokemon, so Battle Tower and others that cannot give EXP cannot give EVs. Neither value changes the other. Whether you get the lowest or highest amount of EXP possible, the EV gain will be constant depending on the fainted Pokemon. And when using vitamins, EXP is not affected, same vice versa with EXP candy. However after gen V, even if you reach level 100 but faint a Pokemon anyways, you get EVs anyways assuming you would get EXP if it wasn't level 100.
Source: https://bulbapedia.bulbagarden.net/wiki/Effort_values does not imply any connection other than what I have mentioned.