I still do not see why this would seem to be such a large disagreement among Presbyterian and Baptists then, as both sides do not see the rite granting saving grace, and both see only saved as being really in the New Covenant now.
Something that seems to be left by the wayside today is the Regulative Principle of Worship (RPW) and related issues when it comes to this question. Maybe this is because you are in the North and these kinds of things (insistence on immersion, restricted communion, etc.) have persisted as emphases among Baptists in the South moreso than in the North, from what I understand. Also, maybe it is less prominent today now that so many Presbyterians have abandoned aspects of the RPW as well, with widespread acceptance of things like "pictures of Christ" among them.
Traditionally, Baptists have seen baptism as immersion, period, especially in the USA. (This one argument may have convinced more people to join Baptist churches in the 19th Century and afterward than any other.) So as AMR said, they effectively unchurch all paedobaptist churches, particularly those Baptist churches who have maintained close communion, which would bar all unimmersed persons from the table, no matter how godly and renowned. If baptism is considered a mark of the church, then I don't see how the Baptist can affirm pedobaptist churches as true churches. (There are some in the Dutch tradition, such as some in the URC that I am aware of, who do not consider Baptist churches to be true churches. But I've rarely if ever seen that alleged by Presbyterians.)
Also, infant baptism is viewed by traditional Baptists (including 1689ers) as a violation of the RPW, an example of "will-worship", something that cannot be demonstrated in the NT by command, example or (legitimate) inference. Representative of this argument is a work by John Gill entitled "Infant Baptism: A Part and Pillar of Popery." (If it's not in the NT, and Rome does it, then....)
Obviously these points are debated by our Presbyterian friends. In other words, this is as serious an issue for Baptists (or it should be) as "pictures of Christ" is for those holding to the traditional Reformed view of the 2nd Commandment or singing uninspired hymns is for the Exclusive Psalmodist.
That you would ask why it is such a big deal after being on this board so long is baffling, frankly. Maybe part of that is because these debates on the PB used to generate a lot more heat 10-15 years ago. There also seem to have been more Baptists here back then.
I think a few decades ago if you asked a Baptist (Calvinist or not) why he is a Baptist, he'd have said "Baptism is immersion" and "Infant baptism isn't in the NT" and generally not felt that a whole lot of argument beyond that was necessary. Also, perhaps before the revival of Calvinistic soteriology in the 2nd half of the 20th Century, there was somewhat less of a felt need among Baptists to be accepted as "Reformed" (or respectable?) by Presbyterians. At the risk of coming across as anti-intellectual, it seems to me that now we are getting all of these Baptist arguments about the NC that are almost as complicated as the Presbyterian arguments, if not moreso.
One book that came out a few years ago that is basically "1689 Federalist" almost "represbyterianized" me with several of its arguments! You end up with over-arguments that to varying degrees can be proved to be false, such as "Circumcision had no spiritual import whatsoever and was only a national symbol" and "Israel only existed to pave the way for Christ" and "Everyone in Israel was unregenerate except for a few prophets and others." Some seem to want to make everything in the OT into a type.
Another factor is the legitimacy of the Presbyterian "two stage" membership with non-communicant covenant children who aren't admitted to the Lord's Supper until roughly the point that the Baptist would baptize them and (more or less) simultaneously admit them into full membership. (Thus, Presbyterians and others who charge Baptists with "withholding the privileges of church membership" to their children are open to the charge that they withhold full privileges themselves. David's "What's the difference" question may have some applicability here.) I think the Presbyterian would say this is necessary via "good and necessary consequence" and cite 1 Cor 11 and maybe a few other texts on proper preparation for the Lord's Supper. The Baptist of course sees this as another example of an unbiblical practice but is thankful that at least most Presbyterians don't practice infant communion along with infant baptism.