Perhaps you have read John Piper's excellent essay, Are There Two Wills in God?. If you have not, I highly recommend it since I will be restating parts of it and building on other parts of it here.
I take as my starting place that both monergists (as usual, the term I prefer to Calvinist) and synergists (in place of Arminians) believe that there are two wills in God, a revealed will and a secret will. (If you are a synergist and you don't think you believe God has two wills, please read Piper's essay. I will be supporting the point more below, but digesting that essay is very helpful for clear thinking on the subject.) Sometimes these are called prescriptive will and eternal will.
Nevertheless, I believe that there is further agreement in that both monergists and synergists agree on God's revealed will: that God loves all men, wishes (in some sense) for them to be saved, and that He commands us to share the Gospel to all men and invite them to repent, believe, and be saved. He also reveals that anybody who hears the Gospel and believes will be saved. Both monergists and synergists agree on these points, since they are clearly laid out in the Bible.
The difference only occurs in what God's secret will is, that is, the basis on which people are saved in God's eternal plan. This difference is because the Bible does not lay this out in a systematic fashion: you have to piece together evidence from many different Bible verses, and people differ on the interpretation. However, again there is some agreement. Both agree (I hope) that God's secret will is primarily to give Himself glory. It is true, synergists do not normally talk in terms of God's glory: they typically talk, like C. S. Lewis did (definitely a synergist, a pretty pure Arminian) about God's upholding men's free will because He did not want an army of puppets. But I hope that the underlying unstated reason is that they believe giving men ultimate control over their own salvation brings God glory in some way.
The crux of the difference between monergists and synergists is what it is that gives God the most glory. The monergist says it is that He is demonstrated to be both merciful (through those He elects) and just (through those He reprobates). For the current purpose, I don't think it matters so much whether God actively reprobates the lost, or simply allows them to perish in their sins: the point is that God could have saved them, but He doesn't, and the fact that He saves some and not others brings Him glory. After all, better knowledge of God brings Him glory, and understanding that he elects some and not others is better knowledge of Him. This is true even for those who are lost: though their will continues to rebel against God, at least the flag of truth is planted within them (as Lewis also noted), even if they don't do anything else with it.
On the other hand, the synergist says that God gets the most glory by allowing man his free will when it comes to salvation. And I can understand that, in a way. Why shouldn't it give God glory? After all, it gives men glory (in their own way) when they respect the free will of others. The problem here is really the same one I mentioned in my last post about the "Free Will Song": many of the standards that apply in human-to-human relationships do not apply in God-to-human ones. It is wrong for a human being to demand worship from another: not so between men and God. One human should not be a slave to another: but Paul said that he was a slave to Jesus Christ.
Similarly, men should give each other free will: but God's understanding is so much above ours the same does not apply to God, and in two ways. He is infinite, of course, while we are finite. But just as importantly, we are sinful and He is not. Our wills are free, but they are enslaved to their own sin. Thus, we neither can nor wish to respond in faith to God. He could have let us go according to our own wills: but then nobody could be saved. It's only because He does NOT let us go according to our own wills that any of us are saved.
I should mention a point once made by a synergist I was talking to (a very intelligent young man, very knowledgeable especially given that he was only 17 years old). He agreed with me that he also believed in the idea of two wills in God; but then he made the comment "But in my understanding the two wills do not contradict each other." We didn't get a chance to hash that out, but I have thought about his position. I think he was saying that in the situation of the people that God reprobates, His two wills are in conflict with each other: the revealed will saying "save the person" and the secret will saying "condemn the person."
It was an interesting comment. It is true, one will says "save" and the other says "don't save." But if the results weren't different at least in some cases, there would really be no reason to speak of there being two wills: there would only be one will. Still, I think it fails as a support for synergism, and that in two ways. First, the two are not really in conflict at a high level: God's revealed will is to save, and His secret will is to bring Himself glory. They are really dealing with two different issues. The only way they can be said to be in conflict is with regards to that particular person.
Second, doesn't the synergist have the same problem? In the synergist's view, God wants to save the person (revealed will), but not wanting to save the person so as not to interfere with his free will (secret will). Again, we have an apparent conflict as regards that one individual person. So if the monergist's version of God's two wills has a "contradiction," so does the synergist's.