Indeed, there is some difference between appear and seem - see, for example, section 5.5. in The Longman Grammar of Spoken and Written English. I'm not going to write about morphosyntactic differences between 'appear' and 'seem' (can't seem, seem like etc.)- you can easily find it in any good dictionary or usage guide, e.g. ldoceonline or dictionary.cambridge.org or learnersdictionary.com.
I'll comment on the semantic difference between these verbs, obviously excluding appear used in the sense of come into existence; come into sight (as in "He appeared out of nowhere."). The authors of the MW Dictionary of Synonyms acknowledge that these verbs are often used interchangeably with "no apparent difference in meaning." However, they remark that "even in such phrases seem suggests an opinion based on subjective impressions and personal reaction rather than objective signs." The verb appear, on the other hand, they argue, may imply "that the opinion is based on a general visual impression" (like the verb look) but it "sometimes suggests a distorted impression such as can be produced by an optical illusion, a restricted point of view, or another's dissembling" (p. 719).
The authors of the Oxford Learner's Thesaurus put it really nicely:
Seem is often used to make what you say about your thoughts, feelings or actions less forceful or to suggest that sth is true when you are not certain or when you want to be polite.
Appear can also suggest that you, or the person you are speaking to, does not quite believe that sb/sth really is as they seem, as in It would appear that this was a major problem (=although I don't really understand why it should be). It can also be used, like seem, when you are not certain about sth or don't want to accuse sb too directly of doing sth. wrong.
Apresjan et al. 1979 argue that the difference between ‘appear’ and ‘seem’ is in where uncertainty comes from. With ‘seem’, that uncertainty comes from the observer/experiencer. With ‘appear’, that uncertainty is caused by the properties/traits of the observed person or thing, and may imply an attempt to deceive.
For example, “tried to appear” used with an adjective is much more common than “tried to seem”:
BNC
tried to appear ADJ – 7 instances
tried to seem ADJ – 0 instances
Anne tried to appear cheerful when she said goodnight to her mother but she was glad to go to bed to try to sort out her thoughts. (G16 A nest of singing birds. Murphy, E. London: Headline Book Publishing plc, 1993.)
COCA
tried to appear ADJ – 35 instances
tried to seem ADJ – 8 instances
or have a look at Google Ngrams
Apresjan et al. 1979 also argue that in cases when you deal with an object or a situation (not a person) and when your impression of that object or situation is based on their internal characteristics (i.e. not their physical properties that can be perceived by our senses), the above-described semantic difference is neutralized.
Practically all well-researched dictionaries and corpus-based grammars mention that appear is less common than seem and that appear sounds (slightly) more formal, cf. the following note in the Longman Dictionary of Contemporary English:
Register In written English, people often prefer to use appear rather than seem, because it is more formal: It appears that the man had been murdered.
As a side note, not being a big fan of prescriptivism, I'd like to quote Bryan Garner:
"The phrase it would appear is invariably inferior to it appears or it seems."
Why he thinks that is a mystery to me.
P.S. The most comprehensive explanation I've ever seen - it takes almost four pages, much more detailed than the one in the MW Dictionary of Synonyms - can be found here (caveat: it's in Russian)
