I am aware that this is an old question, been discussed
Having "due to" is more adjectival, although I am not that particular, but what if the word in front is adjective?
e.g.
Most of grammar books describe that due to acts more adjectival. We all know but this question (+1, of course) forces me to dig in deeper. And, I found something useful...
In the book Woe Is I The Grammarphobe’s Guide to Better English in Plain English, the author proposes to substitute "due to" for “caused by” or “resulting from.” She explains that if a sentence begins with “due to,” as in “Due to inclement weather, school was canceled,” the sentence is “probably wrong.” - Grammar Girl
While keeping general rules aside for a special case like this (adjective ahead), I think following this rule does not harm the structure.
Having said that,
Climbing that mountain is difficult because of its height -sounds preferable to me.
Note - COCAE shows results of both the usages (...difficult because of... and ...difficult due to ...) but then the former returns with over hundred results, the latter sticks around a couple of dozens.
both usage is right in your sentences.
to elaborate the difference, the usage of "due to" is found in more formal / legal situations.
e.g. 1) I got delayed because of traffic (not due to) 2) The cash payable due to him broke me. 3) I like ice cream because of its chillness. 4) His emotions were twisted due to Shakespearean plays.
The phrase "Due to" can, in a very specific context, refer to the holder of a debt. Consider the following example:
"The money due to the bank had not been paid."
It is not at all uncommon, however, to see the phrase "Due to" used in a "because" context, such as the following:
"Due to circumstances beyond our control, today's meeting has been cancelled."
