I have the idea that 'dichotomous' means conflicting and 'dilemma' is state of mind when one/many cannot simply decide between two options. But I'm unable to wrap my head around the phrase. Also if I've made any errors please rectify but do apprise me of them in comments :).
'Dichotomous' comes from dichotomy and could be 'two-fold', 'different from each other', or 'mutually exclusive'. The latter is often the case in science where you have dichotomous variables or dichotomous factors.
DICHOTOMOUS comes from 'dichotomy', which means:
Cambridge Advanced Learners Dictionary
a difference between two completely opposite ideas or things;
There is often a dichotomy between what politicians say and what they do.
From Oxford Advanced Learner's Dictionary 8th Ed
the separation that exists between two groups or things that are completely opposite to and different from each other
From Longman Dictionary of Contemporary English 5th Edition
the difference between two things or ideas that are completely opposite
a situation in which a difficult choice has to be made between two different things you could do
Example 1: The President is clearly in a dilemma about/over how to tackle the crisis.
Example 2: She faces the dilemma of disobeying her father or losing the man she loves.
There are two or eventually more mutually exclusive (dichotomous) options available and to make a choice is very difficult (dilemma).
Since the word 'dilemma' has the connotation that it's a choice between two things, the expression
dichotomous quandary might be more appropriate in case that there be more than 2 options. Just an idea of mine.
To me personally, a dichotomous dilemma would indicate a situation of foremostly two options and no matter which choice one takes, there is always an ensuing 'loss' or 'downside' once the choice has been made. Dichotomous, by some considered tautological in this example, does stress the ordeal of the dilemma and the very different options/results. ...Other forum participants might have a different option.