@ eipi10 answers this particular case, but overall you also need to align the width of the graph. If, for example, the labels y on one of the graphs take up more space than on the other, even if you use the same axis on each graph, they will not line up when transferred to grid.arrange:
axis <- scale_x_continuous(limits=range(m$x))
Line_plot <- ggplot(data = m, aes(x = x, y = y)) + theme_bw() + axis + geom_line()
m2 <- within(m, y <- y * 1e7)
Bar_plot <- ggplot(data = m2, aes(x = x, y = y)) + theme_bw() + axis + geom_bar(stat = "identity")
grid.arrange(Line_plot, Bar_plot)

In this case, you need to align the width of the graph:
Line_plot <- ggplot_gtable(ggplot_build(Line_plot))
Bar_plot <- ggplot_gtable(ggplot_build(Bar_plot))
Bar_plot$widths <-Line_plot$widths
grid.arrange(Line_plot, Bar_plot)

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