In terms of meaning, there is no difference between the second and third sentences.
It seems that they have not completed the task yet.
It seems they haven't completed the task yet.
The word that is not necessary to be there when it is used as a conjunction to introduce a subordinate clause expressing a statement or hypothesis.
I hope that I can fly.
I hope I can fly.
As suggested by RegDwighт♦, you can read more about it here.
If you care a lot about the details,
there is actually a very small difference between the first and second/third sentences.
It seems like they have not completed the task yet.
It seems (that) they have not completed the task yet.
The word like here is used as a preposition:
like
having the same characteristics or qualities as; similar to
and the interesting thing is, the word seem has the definition:
seem
give the impression or sensation of being something or having a particular quality
So when you use the second/third sentence
It seems (that) they have not completed the task yet.
you are giving a hypothesis as you are saying that
It gives an impression that they have not completed the task yet.
which shows that you are not totally sure if they have not completed the task.
Using the sentence with the word like would indicate that you are even less sure if they have not completed the task.
It gives an impression that is similar to that they have not completed the task.
Yes, there is a name for this kind of alternation between constructions.
It's called Negative-Raising, or Neg-Raising (NR), among other things,
and it's governed by the predicate seem in this case;
there are a number of other predicates that govern it.
NR is a minor cyclic alternation rule.
That means that it is governed by the matrix predicate (all cyclic rules are governed)
and that the set of predicates governing it is small and specialized (that's the "minor" part)
and that it relates two different but synonymous sentence structures (that's the "alternation" part).
What happens is that, when you have a complement clause with a negative in it, like
- Bill wanted/seemed/intended/tried/managed not to be driving the truck.
with some predicates, but not others, this construction is equivalent to the same
sentence with the matrix predicate negated, but the complement not negated:
- Bill didn't want/seem/intend to be driving the truck. (equivalent with want, seem, intend)
- Bill didn't try/manage to drive the truck. (not equivalent with try or manage)
I.e, want, seem, and intend govern NR, and try and manage don't govern it.
Another way to look at it is that NR predicates are "transparent to negation", because
they don't really contribute much to meaning beyond individual perceptions and desires.
Whereas the vast majority of complement-taking predicates do contribute to meaning,
and are therefore "opaque to negation".
By the way, I used infinitives in the examples above for simplicity, but the phenomenon
is not limited to them. With the right predicates, tensed complements can undergo NR, too:
- I thought (that) you didn't want toast. = I didn't think (that) you wanted toast.
but ...
- He said (that) you didn't want toast. ≠ He didn't say (that) you wanted toast.
Best Answer
No, there is no difference.
Seem is a verb that governs infinitive complements and allows Negative-Raising. That means that negation in the infinitive complement of seem, or want, or other Neg-Raising verbs, as in
can also appear, instead, in the matrix clause with seem or want
without a change in meaning.
This is not true of most predicates, which don't allow Neg-Raising. (Be) Easy, for instance, is a more normal predicate; the two sentences below do not mean the same thing.