Where they are synonyms in some cases (and can be used interchangeably without changing the intended or inferred meaning), there can once be a slight difference: I

1. saw John at the hospital and he appears fine.
2. John is in a intensive care unit. I see him three times a day. I find he is all right.

No (1) is usually straightforward and positive, while No (2) may carry the additional meaning of the speaker being less sure of John’s condition, in that outwardly John “appears” okay, but that may not be the whole story.

How do the words are stressed (when spoken)?

To generalise: if there is the need to show that there could be any doubt at all (even if only slight), then use appears rather than appears.