Why is difficulty used here(and not diffucult). What part of speech is it?
There's an ellipsis of a preposition before "breathing":
"With panic disorder, a person has brief attacks of intense terror and apprehension, often marked by difficulty [in] breathing."
"Difficulty" is a noun functioning as the object of the preposition "by."
"Breathing" is a noun (a participial noun) functioning as the object of the omitted (but understood) preposition "in."
To those averring the following:
"The utterer of 'I have difficulty breathing' is not silently thinking 'in'."
How do you know that? In any case, I'm neither a mind-reader nor a psychologist, and it makes no difference at all what was occurring in the mind of the person who made the utterance. My job is to parse the construction according to consistent principles of syntax, and not to invent new categories, ad hoc, for the sake of easy explanation.
So "X had difficulty [in] breathing" is a correct parsing of the construction, irrespective of whether or not the originator of the utterance was "silently" thinking of the preposition. I don't know what he silently thought; neither does anyone else except the utterer. Constructions assume a force of their own even if someone who utters them did not intend it, or never thought, it or even denies it later on. "All professors are absent-minded, and Joe is a professor . . . oh, but, hey, I'm certainly not thinking that Joe is absent-minded!" Right. It matters not. The innuendo of the unstated but understood conclusion presented in the enthymeme is ineluctable.