An earlier version of this answer was originally posted as an answer to the question “Consequences of Unique Games being a NPI problem” by NicosM. Because it turned out that it did not answer what he wanted to ask, I moved it to this question.
Short answer: They mean different statements. The latter implies the former, but the former does not necessarily imply the latter.
Long answer: Recall that the unique game problem is the following promise problem.
Unique game problem with parameters k∈ℕ and ε, δ > 0 (1−ε > δ)
Instance: A two-player one-round unique game G with label size k.
Yes-promise: G has value at least 1−ε.
No-promise: G has value at most δ.
The unique games conjecture states:
Unique games conjecture. For all constants ε and δ, there exists a constant k such that the unique game problem with parameters k, ε, and δ is NP-complete.
Consider results of the following form:
(1) Assuming the unique games conjecture, problem X is NP-hard.
(An example of X is the problem of approximating maximum cut within some constant factor R > RGW.)
Most (if not all) of the results of the form (1) actually prove the following fact:
(2) There exist constants ε and δ such that for every constant k, the unique game problem with parameters k, ε, and δ is reducible to X.
It is easy to verify that (2) implies (1). However, (2) implies more than (1): for example, suppose that one day we can prove that a variant of the unique games conjecture where “NP-complete” is replaced with “GI-hard.” Then (2) implies that X is also GI-hard. (1) does not imply this. This is why some people consider that statement (1) is not the best way to state the theorem: (1) is weaker than what is actually proved, and the difference might be important.
Although (2) is a more accurate statement of what is proved, it is clearly mouthful. This is why people have come up with a shorthand for it: “Problem X is UG-hard” is a shorthand for (2).