snap to grrid will not work for drawing bezier curves. Why not?

Snapping has become quite a complex feature, over the last few years. To snap lines drawn with Pen/Bezier or Pencil/Freehand tools (both draw paths) to a grid, you'll need to 1 - engage the following buttons, *AND* 2 - disengage all the rest.

Enable snapping (1st button at top of snap control bar)
Snap nodes or handles
Snap to paths
Snap to cusp nodes
Snap to smooth nodes
Snap to grids

And there are a few other buttons in the Snap nodes or handles section of the snap control bar, which might or might not apply to your drawing, and what you want to accomplish. With the info you've provided, I have no way of knowing which you might or might not need.

Also go to Document Properties > Snap. This is a general setup that should help you control the snapping. Although you may need to customize it, as you gain more experience. For each of the 3 sections (Snap to Objects, Snap to Grids, and Snap to Guides, choose "Snap only when closer than" and set the slider in the middle (20 to 40). If you set something to "Always snap" everything in sight seems to want to snap, which can be frustrating. I find that to be a good general setup, so that I don't always have to come back and tweak the settings for each image I'm working on. Although everyone develops their own personal style, and it might not annoy you as much as it annoys me, to be constantly changing those Doc Props settings. :-)

And for a more detailed explanation of Snapping, be sure to see the manual: http://tavmjong.free.fr/INKSCAPE/MANUAL/html/Snapping.html