When someone says "something appears to be broken", does that mean he is assuming that it might be broken or is he certain that the said thing is broken?
appears to be and seems to be both carry the same meaning - that something gives the impression of being a certain way. It's not a simple assumption - it implies a high level of probability, while still allowing for a margin of error.
In practice, appears to be and seems to be are both used to state something as fact, while providing a sense of distance or insulation from having to stand by that fact.
The child seems to be healthy, but the doctor is concerned.
The house appears to be in good condition.
Both can also be used to convey an element of surprise, while still stating something as fact.
There appears to be an error with my bill!