My understanding is that "seem" has the connotation of sense or feeling whereas "apparent" is a slightly harder realization. Seem is vague perception while apparent is slightly less vague perception. Take this sentence for example. "She left the room seemingly all right, but that apparently was not the case when she was heard sobbing outside the door." If "seemingly" and "apparently" were interchangeable, you could switch their positions in the sentence and the sentence would still have the same meaning. However, if you switch the words, the second half of the sentence makes no sense at all because there is no vague perception of what has happened. It is like the difference in water at 0 degree centigrade and 1 degree centigrade. One is frozen and the other is not quite frozen. I hope this helps.

While the two are very similar, and can often be interchanged without changing the meaning of a sentence, there are certain occasions where one wouldn't be used for the other.