Learn English – What’s the difference between “seem” and “seem as if”

They do not differ significantly in meaning.

The relevant meaning of seem is this one from the Oxford English Dictionary:

3. With noun, adj., or phr. as complement: to appear to be, to be apparently (what is expressed by the complement).

Under this meaning there are three distinct submeanings. The first is the simple case, corresponding to your first example. The second suggests usages involving the perceiver being brought in following "to," like "the children seemed to me ..." The third corresponds to your second example:

c. Followed by †as, as if, as though.

There's no suggestion of a difference of meaning. Indeed, for both usages, they are understood to say that the children appear to be tired. In either case the appearance could deviate from the fact of whether they are tired.

Furthermore adding "as if" does not indicate an imaginary situation, since as a conjunction the phrase often means (via Wiktionary):

As though; in a manner suggesting.

In the form "seemed as if," the usage is redundant (pleonastic) but nonetheless idiomatic: the children appear tired; they appear in a manner suggesting they are tired. While it would be tempting to read the phrase with the second meaning of "as if" ("in mimicry of" - the children act as if they are tired, but they are mimicking tiredness), that would be an error, since it doesn't catch the idiomatic interpretation of "seemed as if" captured in the dictionary entry for seem.

So the only question is a stylistic one: which do you like more in a given situation? Are you being concise, or is there something you value in redundancy that makes your expression harder to miss?