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 inherently 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.
AF "points" are actually a pair of lines that can detect contrast when the lines of that contrast are at high angles to the orientation of the lines. A cross type focus point is actually two regular AF points arranged at 90° to each other.
Faster and/or more accurate depends on several variables.
The shot to shot variation of the camera/lens focus system. When it was first introduced, PDAF was designed to be fast at the expense of absolute accuracy. As the technology has matured it has gotten more accurate, but speed is still a big part of designing and evaluating a PDAF system. While we would all like for PDAF systems to be instantly fast and perfectly accurate, in practice most systems have to balance speed and accuracy against each other. How a system is designed and how the firmware that runs it is written will determine the balance between accuracy and speed in PDAF systems. Some upper tier bodies even allow the user to change settings that vary the weight of one against the other.
The design of the Phase Detection Auto Focus (PDAF) array. Focus 'points' are really pairs of lines that measure the light falling on opposite sides of a lens. The further apart each line in the pair is, the more accurate that pair can be, but only when the maximum aperture of the lens allows light from that width to reach the focus array.
The maximum aperture of the lens being used. Many cross type focus points will not function at all if the maximum aperture of the lens is too narrow. The same is true of single orientation line pairs, but those tend to be not as widely separated, and thus sensitive at smaller apertures, than most cross type points in a typical PDAF system. The tradeoff with narrower line pairs is that they are less accurate than longer line pairs.
The amount and orientation of contrast in the target of the focus system. This is the most significant variable in terms of the advantages of cross type focus points. If the lines of contrast on your target run the same direction as a single set of lines on the focus array, the AF system will have a hard time focusing on the target. In fact, it may not be able to focus at all. Most AF systems that use non-cross type points place some of them in horizontal orientation and others in vertical orientation. When using such a camera, it is helpful to remember which points are vertical and which are horizontal, and then use an appropriate point when your target has strong contrast lines only running in one direction. There have even been a few designs in the past that used a single pair of lines that were diagonal, so that they would be 45° from both horizontal and vertical lines in the target.
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 horizontal 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 focus 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.