Dilemma

A dilemma ^[1] is when you must must make a choice between two options, and both have nasty consequences.

For instance, in Roald Dahl ^[2]’s James and the Giant Peach ^[3], James and the insects are inside a giant peach that is floating in the ocean.  Their dilemma is:
 
Do they eat the peach, and face death by drowning?

OR

Do they not eat the peach (keep it as a floating vessel), and face death by starvation?

Conundrum

A conundrum ^[4] is a puzzling or difficult problem; almost everybody knows the most famous conundrum of all:

Which came first?  The chicken or the egg?
 
*Note:  ”dilemma” is often misspelt as “dilemna”