grid.arrange draws directly on a device. arrangeGrob, on the other hand, doesn’t draw anything but returns a grob g, that you can pass to ggsave(file="whatever.pdf", g).
The reason it works differently than with ggplot objects, where by default the last plot is being saved if not specified, is that ggplot2 invisibly keeps track of the latest plot, and I don’t think grid.arrange should mess with this counter private to the package.