1) Although both tools are useful, I agree with Fasih that Quizlet is the better of the two for students. I tend to emphasise the need for students to constantly learn vocab and Quizlet is a great tool for that task. I have used other flashcard apps in the past, but Quizlet has more features (including the clincher for me, which is the option for the student to play the word and heat the correct pronunciation).
2) I would say that Socratic is better for in-class. I would usually set vocab acquisition as homework (especially because different students know different words) so I would encourage Quizlet use as individual. Having said that, is it possible to create a Quizlet that students could collaborate on and all add their own words? A class Quizlet could be a great focal point for vocabulary the whole class need. I recently did a session on linkers and it could have been fun to get students to create sets for each other (although this would have probably been a homework). Socratic could be used in any lesson I feel, but perhaps it would be a nice way to make a reading comprehension more interactive.
3/4) Maybe games and trying to make things more fun can give a slightly wrong impression to EAP students. The stakes are high for many and there is a lot of work to do, so some students might feel that messing about with apps is wasting valuable time.