Do cross type focus points achieve sharper photos than horizontal points or do they just grab focus faster.
They are capable of focusing on lines in two directions, so they will handle more situations. Focus points are generally either able to focus on strong horizontal lines or strong vertical lines, cross type can handle both, and thus can handle more scenes well.
If you had only vertical lines and tried using a vertical focus point(optimized for horizontal lines) it would have a hard time, but if you had horizontal lines, you wouldn't notice a significant difference between the vertical focus point and the cross type as they both can ideally handle the situation.
The advantage of Cross Type Focus Points isn't that they are faster, more accurate, or both: it is that they can successfully deal with a wider variety of situations than focus points with only a single orientation can.
Faster and/or more accurate depends on several variables.
Here's a map of the Canon 5D mark II focus system. The focus points visible in the viewfinder are the small black rectangles. The areas of sensitivity for each 'point' are shown by the blue rectangles. Notice that only the center point is sensitive in both the vertical and horizontal directions, the rest are either vertical or horizintal only. The red points are smaller assist points for the center point when the AI Servo option is selected. These points help the system track moving objects in the frame.
Here's a map of the more complex system in the Canon 7D, along with a diagram of the focus array sensor and a chart that tells which sets of lines are responsible for which focus points. Notice that all of the focous points are cross type, and the center point includes a diagonal cross type point as well. If you look at the chart of the sensor array, you see that the lines for the diagonal cross are spread further away from each other, and thus more accurate, but only usable when a lens with an aperture of f/2.8 or wider is used.