Traditional Reformed sacramental teaching is indeed a lot more “Catholic” than many folks realize (in fact, it’s more Catholic than many Presbyterians realize). The basic difference between traditional Protestant sacramental theology and Catholic teaching is usually stated this way:
Protestants believe that you have to have faith (faith-that-works-by-love, bear in mind, not just what Catholics call “faith”) in order to receive grace from a sacrament.
Catholics believe that the sacraments confer grace to anyone who doesn’t put an obstacle in its way.
So wait a minute: wouldn’t lack of faith be an obstacle according to Catholic teaching? Yes, of course it would. If a person refuses to believe, the sacrament won’t give them any grace. So in part the difference seems to be one of definition and formulation–no doubt there are substantive theological issues here, but they’re highly nuanced ones.
The big difference between Reformed and Catholic sacramental teaching, as Itwin said, has to do with predestination. Specifically, the Reformed believe in “perseverance of the saints,” which means that all those who once receive the grace of regeneration will persevere to the end and be saved. Or, in more negative terms, only the elect ever experience regeneration. Since neither Catholics nor Protestants believe that baptism guarantees final salvation, it follows that baptism does not necessarily regenerate. To be precise, an elect person also may not experience regeneration when baptized–it’s up to God, and God might not choose to give the person that grace until later. The Reformed believe that when God regenerates someone, that person will always believe, and will always persevere. Catholics don’t believe this.
Practically speaking, at least when the WCF was written, the main difference pertained to infants. A regenerate Protestant, and a Catholic in a state of grace, look pretty much the same for the most part, with the only real difference being that the Protestant’s condition is permanent whereas the Catholic’s condition can fluctuate. But if you believe, as Catholics traditionally did, that unbaptized infants could not experience the beatific vision, then there’s a real difference in how the two traditions understand the significance of baptism for infants. That difference, of course, has more or less vanished in more recent times, with the clarification by the Church that we can indeed hope for the salvation of unbaptized infants.
I think there remains a basic difference in how the physical and spiritual elements of the sacraments are related to each other. The Reformed tradition is very afraid of idolatry, and their understanding of idolatry is, in my opinion, too influenced by a certain kind of Neo-Platonism used as a lens through which to understand the Old Testament. So they worry a lot about giving the physical elements in Christianity too much prominence. They want to stress the priority of the spiritual. So while one can argue that in the end the substantive differences are very nuanced (since both traditions believe that grace always comes along with faith that works through love, and both traditions believe that God uses the sacraments to instill living faith in people and nourish that faith when it’s there), the rhetorical differences are big, and this has big effects on the kind of spirituality that results.
Reformed can’t affirm traditional, Catholic teaching about the sacraments without sticking in all the conditions and exceptions and qualifications. Catholics may grant lots of conditions, but those don’t shape the basic way the sacraments are spoken of.
So the Reformed view would be: “God offers regeneration through baptism, and those whom God sovereignly moves receive it,” the Catholic view would be, “God regenerates through baptism, but unbelief and/or stubborn adherence to sin prevent a person from receiving regenerating grace, or cause him/her to lose the grace once received.”
A final, nuanced point about baptism particularly: since it’s the sacrament of the beginning of faith, and since the Reformed believe that regeneration precedes faith, it isn’t actually correct to say that the Reformed think you have to have faith in order to be regenerate. But it is correct to say that those who are not sovereignly regenerated by God will fail to believe even as they are being baptized. Again, the order rather than the final result is different, but that’s still significant.
Most later Protestants (including many Presbyterians and other Reformed folks, especially in America) took the suspicion of physical means of grace much further than the WCF does, and read the WCF through that lens. So you’re right–this does sound quite different from what you would hear from most American Protestants today.