The difference is that the non-presentist (e.g. eternalist) will accept 'Socrates exists now' in one sense but not the other; she will accept it in the ontological sense but not the temporal location sense. For she does not believe that Socrates is present (temporal location sense), only that he exists to be quantified over or referred to (ontological sense).
A singular proposition (according to Markosian and Robert Adams) about x is one that refers to x. That makes being about an individual the same as referring to an individual, which one might reject for a number of reasons: e.g., one might be a presentist who holds that 'Socrates was wise' is about Socrates but refers to nobody. At any rate, the problem is that there are apparently singular propositions about non-present things, such as Socrates. Yet presentism entails there aren't.
According to special relativity (SR), there is no notion of absolute simultaneity, so what is present depends on a frame of reference: something might presently exist according to my frame of reference but not to something else's frame of reference. Since presentism is a thesis about what exists, and since we don't want to say that what is exists is relative (e.g. to a frame of reference), it looks like we have to deny presentism (assuming SR).
Haecceities are individual thisnesses, an example being the property of being identical to Angela Merkel. They are properties that, if they are instantiated, they are instantiated by precisely one thing. The presentist could solve the argument from singular propositions by saying that a sentence like 'Socrates was wise' expresses a proposition about, or that refers to, a Socrates haecceity. So singular propositions about non-present things do exist, it's just that they don't refer to concrete things like Socrates but special properties called haecceities.
We say that e.g. 'Socrates was wise' expresses the proposition that it was the case that 'Socrates is wise' is true (or that something "Socratizes" and is wise) as soon as Socrates goes out of existence, and while Socrates exists it expresses a genuine singular proposition. What is odd about this view that Markosian doesn't mention is that it implies that many of Glaucon's beliefs about Socrates are either lost when Socrates goes out of existence, or they "change", i.e. he loses the old singular proposition ones and gains new paraphrase ones in their place.
He claims that it is not a genuine relation, and gives a paraphrase strategy as follows. To say that Ned admires Socrates is to say that there are some properties that Ned associates with 'Socrates' and that evoke in Ned the feeling of admiration, and there was something that has those properties that is also the referent of 'Socrates'. But now what about a sentence like 'Socrates walked over such and such an area'? It seems that the paraphrase strategy won't work since no intentional notion is involved (such as admiration).
Roughly, the linguistic meaning of a sentence is its truth condition; e.g., 'Jane is happy' is true under the condition that there is a referent x of 'Jane' and x is happy. and that condition is the linguistic meaning of the sentence. The propositional content of a sentence (token) is whatever is expressed by the sentence so as to make it true or false. This depends on one's theory of propositions, but if we suppose a proposition is a sequence of things referred to by the sentence (a so-called Russellian proposition), then the content of 'Jane is happy' is the sequence (Jane, happiness) whose first member is Jane and second member the property happiness, and it is true just in case the first member has the property that is the second member.
What Markosian says is that, even though a sentence like 'Socrates was wise' has no content, it does not follow counterintuitively that it is meaningless--for it still has linguistic meaning (truth/falsity conditions).
He says that 'Socrates was wise' is true iff there is a referent x of 'Socrates' and it was the case that x is wise. Given this truth condition, the sentence is--counterintuitively--untrue since there is no referent of 'Socrates'. (Further, it is not false assuming, as Markosian does implicitly, that the falsity condition is analogous to the truth condition--there must be a referent of x that was such that it is not wise. As there is no such referent, the sentence is neither true nor false.)
The distinction rests exclusively on the scope of the tensed operator "was". Grabby ones give "was" so-called narrow scope, while searchy ones give "was" so-called wide scope. Consider e.g. 'The number of planets is necessarily greater than 7'. Giving 'necessarily' wide scope, i.e. reading the sentence as 'Necessarily, the number of planets is greater than 7', the sentence is false since there could have been fewer than 7 planets. Giving it narrow scope, i.e. reading it as 'The number of planets, namely 8, is such that it (namely 8) is necessarily greater than 7', it is true. Similarly, the narrow scope or grabby reading of past-tensed sentences makes them false (such as 'Socrates was wise'), while the wide scope or searchy one makes them true.
Markosian goes for the grabby ones because he thinks a sentence like 'Joe Montana was a quarter', where Joe Montana still exists to this day, refers to Joe Montana now, and would not be true even if there was a different Joe Montana that was a quarterback. As such, the sentence must get grabby truth conditions. He also does not think that the truth condition for the sentence changes from grabby to searchy once Montana goes out of existence.
His argument is that either STR does not entail that there is no notion of absolute simultaneity, in which case premise 2 of the argument fails, or it does entail this, in which case he rejects the theory (hence premise 1) in favor of a weaker one not endorsing this entailment. The reason rejecting such a strong version of STR is that the empriical evidence is compatible with both, and since he has a priori reasons for rejecting the stronger, that leaves only the weaker theory left.
Abstract times (even the now) are maximal, consistent propositions. For every proposition p, (i) a time entails either p or its negation (so it is maximal), and (ii) it never entails both p and its negation (so it is consistent). Abstract worlds are just the same. (I don't like this way of defining worlds, though it is the standard one, since it collapses the distinction between times and worlds, and since it then becomes impossible (given a usual world-making language) to extract the structure of time from a world. Rather, we should let worlds be ordered sets of times. This also answers an objection Markosian mentions.)
This allows the presentist to be able to quantify over and refer to non-present times and to treat tensed operators like "was" and "will" as quantifiers over times, just as is done for modal operators like "possible" that are standardly taken as quantifiers over worlds.