Hey buddy,
they have not much difference except the screen size, mi max is a little big, 6.44" screen the note3 is 5.5" that 1" difference is something you have to consider, if you are ok with big screen go for mi max, otherwise go for note3, note3 is very nice and powerful phone, and the price is less. and with a good ram and rom.I am happy with this phone, good quality, quick and responsive, does any job.