It seems that he is rich.
and
It appears that he is rich.
These two sentences seem to me similar. Is there any slight difference?
It seems that he is rich.
and
It appears that he is rich.
These two sentences seem to me similar. Is there any slight difference?
While they are synonyms in a lot of cases (and can be used interchangeably without changing the intended or inferred meaning), there can be a slight difference:
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.
This difference can be substantially greater, depending on how the words are stressed (when spoken).
To generalise: if there is the need to imply that there could be any doubt at all (even if only slight), then use seems rather than appears.
"seems" and "appears" are synonyms as used in the sentences you provided.
Definition 1 of SEEM according to Merriam Webster
to appear to the observation or understanding
and
Definition 3 of APPEAR according to Merriam Webster
to have an outward aspect : seem