A few general comments:
It is easy, even with little background, to create conjectures. Many of these can either be settled easily or they are considered uninteresting by the culture of mathematics. It is much harder to find a conjecture that is hard to prove and considered interesting by that culture. Ronald Graham and his group have written a program called Graffiti that comes up with conjectures in graph theory, almost all of which are easy to state, almost certainly true and seem far beyond anyone's ability to settle, but thus far with few exceptions the culture seems to consider these to be uninteresting.
Problems that involve addition and primes seem to often be harder to settle. There is a field called Additive Prime Number Theory, but it seems that the tools we have to settle questions involving addition and primes are fewer and weaker.
If you have to reach into a pot and pluck out one prime that satisfies a condition then the primes seem "dense enough" that many conjectures are reasonably easy to determine true or false, with some exceptions. But if you have to reach into the pot and pluck out two numbers at the same time that together satisfy some condition then these seem to be not "dense enough" to be able to settle conjectures as easily, and not so sparce that it is easy to settle the question either.
Both Goldbach and twin primes involve pairs of numbers and the subject of primes and addition. Both are difficult problems that have resisted the best. I think I remember looking at a list of open questions years ago and recognized that a number of them involved pairs of numbers and primes. I don't remember any other example, but I would be interested in seeing any other examples or whether folks here could come up with new problems that fit this description.
Since you specifically asked what the big deal was with this problem, someone writing about the Clay math prizes perhaps a decade ago mentioned that there are lots of math problems that are simple to state and hard to prove, but only a select category of math problems seems to attract the attention of amateurs, the rest are ignored by amateurs.
And finally I recall someone writing about music. They said that hundreds of years after an individual is dead almost all of the music of that time has been completely and totally forgotten, only a tiny fraction that escaped accidental destruction or being discarded by the cultures of the following centuries remains.
So perhaps the reason we remember Goldbach's conjecture today is at least partly influenced by all these components.
