Does “Q” elucidate the "synoptic problem"?

Proponents of the Q theory would say that none of the alternatives provide an adequate explanation of all of the parallel passages in the synoptics. Leaving to the side any of the non-Markan-Priority theories (as they are even more minority views than Q-less views), you're left with basically two options without Q: Matthew used Luke, or Luke used Matthew. Proponents of Q would say there are substantial problems with either model.

One example from An Introduction to the New Testament by Carson and Moo concerns the order of events in the gospels. They give this table on page 89:

The bold passages are those where Matthew or Luke have a different order than Mark. There are times when Matthew and Mark have the same order and Luke is different, and times when Mark and Luke have the same order and Matthew is different, but no times when Matthew and Luke have the same order and Mark is different.

Note that Matthew and Mark agree, against Luke, in placing the accusation that Jesus casts out demons in the name of Beelzebul just before the so-called parables of the kingdom; and Luke and Mark agree, against Matthew, in putting the stilling of the storm and the healing of Gerasene demoniac just after these parables. At no point, however, do Matthew and Luke agree against Mark. To put it another way, at no point does Mark follow an order that disagrees with the other two (hence the lack of any bold type in the Mark column.) (page 92)

If Matthew or Luke used the other, we would expect that there would be occasions when the earlier gospel changed the order from Mark, and then the later gospel used the same order, i.e., there should be times when Matthew and Luke agree against Mark. And you also have the problem that, whichever one you think is final, there are passages where it would appear to have rejected the order of the other and then reinstated the order from Mark. If Matthew is final then you have to explain how it took Luke 11:14-32 and 8:19-21 and repackaged it into Matthew 12:22-50 so that it matches the order of Mark 2:20-35. If Luke is final then you have to explain how it took Matthew 8:18, 23-34, 9:18-26 and repackaged it into Luke 8:22-56 so that it matches the order of Mark 4:35-41. (This option is less problematic because you could say that Luke just ignored Matthew and copied Mark directly. If you want to propose Matthew is final then you have to account for using Mark's order but adding in Luke 11:29-32.) With Q the explanation is simpler: Matthew and Luke used Mark, reordering it at times, but without reference to the other gospel.

For another example, consider the Sermon on the Mount (Matthew 5-7) and the Sermon on the Plains (Luke 6:17-49). There is lots of overlap here, which is not shared with Mark, and yet there are significant differences that would be hard to explain if one of the Gospels had simply used the other. The two texts are easier to explain by them both using a shared source, editing and expanding it to serve the different purposes each had in writing their gospel. (For example Matthew wants to show Jesus's relationship to the Jewish Law far more than Luke does.)

First up, the blessings in Matthew 5:3-12 and Luke 6:20-26. Matthew has a longer passage, with Luke only using half of the blessings. But Luke then has four woes which are not in Matthew. There are also several vocabulary differences so it's not the case that one simply copied the other, leaving out some verses.

Luke's passage on loving your enemies (6:27-36) is similar to Matthew 5:38-48, but there are many differences, and they have similar things in different orders. I've coloured the similar verses myself here. Note that each gospel has places where they are longer than the other. And Luke includes the golden rule here while Matthew has it later in 7:12. There's no simple way to explain one gospel using the other, as it would have to simultaneously condense and expand it. But if Q had only the shared parts I coloured, then it's easier to explain each gospel expanding upon it, no condensing needed.

Another noticeable difference is the passages on judging others: Matthew 7:15 and Luke 6:37-41. Although Luke's Sermon on the Plains is much shorter overall, here Luke is quite a lot longer. Luke's parable of the blind leading the blind is found elsewhere in Matthew 15:14, and the first part of Luke 6:38, "Give, and it will be given to you. A good measure, pressed down, shaken together and running over, will be poured into your lap." is not found anywhere in Matthew.

This page by Peter Kirby covers better than I can many of the arguments for Luke's independence of Matthew, ie, that he didn't know of or have a copy of Matthew when he wrote his gospel.