The author of a famous book on English usage, H.W. Fowler, suggested making the distinction between the two words more clear by contrasting opposing terms to partially and partly in order to orient them:
partially = incompletely = to a limited degree
He has only partially succeeded in his mission.
partly = in part = as regards a part and not the whole
Either partially or partly could plausibly be used in both of the above example sentences, but they illustrate Fowler's suggestion that a distinction might be made in such cases.
However, even though the the words are often used interchangeably, there is some differentiation that we can see by observing usage patterns.
Partially is used more often than partly to modify an adjective or past participle that names or suggests a process:
His face was partially concealed by a beard.
The snow had partially melted.
Our vacation was partially paid for by the company.
Partly is used more often than partially before clauses and phrases offered as explanation:
We trusted him partly because he was elderly.
Partly for this reason, we decided not to buy the house.
I called him again, partly to reassure him.
It must be said that there are plenty of exceptions to this general trend, and only time will tell whether the process of differentiation will continue. None of the above examples would sound incorrect to a native speaker if the words partially and partly were exchanged.
Let's look at the three sentences below. All use partly, but can any of them also use partially?
1. I didn't enjoy the trip very much, partly because of the weather.
2. This he did with difficulty, partly on account of his bad eyesight.
3. It was partly their responsibilty.
We can try to apply Fowler's rule or look at the usage trends, but to ultimately this may become a question of style and not definition or usage. Any of these three sentences could work just as well with partiallyinstead of partly. The learner of English should focus more attention on those few cases that show nearly consistent use of one or the other:
partly cloudy/sunny skies
partially hydrogenated oils