according to AHD they are synonymous:
seem:
1. To give the impression of being; appear: The child seems healthy, but the doctor is concerned. 2. To appear to one's own opinion or mind: I can't seem to get the story straight. 3. To appear to be true, probable, or evident: It seems you object to the plan. It seems like rain. He seems to have worked in sales for several years. 4. To appear to exist: There seems no reason to postpone it.
appear:
1. To become visible: a plane appearing in the sky. 2. To come into existence: New strains of viruses appear periodically. 3. To seem or look to be: appeared unhappy. See synonyms at seem. 4. To seem likely: They will be late, as it appears. 5. To come before the public: has appeared in two plays; appears on the nightly news. 6. Law To present oneself formally before a court as defendant, plaintiff, or counsel.

again, it would depend on context.
English doesn't appear or seem to differentiate in the same way that German does

odondon: Your answer seems comprehensive. Thanks a lot!