They are very similar in meaning in the contexts you have provided, so in my understanding the difference of which verb choice to make is a stylistic one, and has to do with common usage.
"break off" is used when you talk about discontinuing a relationship of some sort, as in "break off diplomatic relations": http://www.merriam-webster.com/dictionary/break%20off In your example, I think "America broke diplomatic relations..." sounds a bit out of place, unless it's being used in a comedic sense (as in causing something to be broken).
Similarly, "to break ties" is an expression in common usage. You could probably say "to break off ties" and it might imply that the ties were broken suddenly.
In general you'll find that phrasal verbs which are synonymous with other verbs differ primarily in the contexts where they are used, so that inserting one of the synonyms wouldn't be technically wrong, but it wouldn't sound "natural".