There are some key differences between a picture style and a white balance setting. While both can affect white and color balance, simply because of the nature of color, the two are intended for different purposes.
Picture Styles affect the baseline curves applied to the image when interpolating the raw bayer pixel information into RGB pixels for viewing on a screen. Picture style curves attune each color channel, introduce stronger contrast over the raw luminence and color information. Its very similar to tweaking the RGB curves in Photoshop, only processing RAW sensor data (individual Red, Green, or Blue bayer pixels) rather than full RGB pixels. The goal is to tune color balance. A classic Color Checker card will help you see the differences between various picture styles, and you can compare the results of a photo of such a card with the card itself or a computerized version (only valid on a properly calibrated computer screen) to see how picture styles affect color balance.
White Balance settings affect the color axis shift of an image. These shifts are primarily along the blue/orange-yellow and green/magenta axes, which are fundamental axes of color theory and human color perception. Most lighting that illuminates a scene is black-body lighting (light produced by dark materials that emit light as a result of being headed to anywhere from 2000 to 20,000 degrees kelvin), so the primary mode of adjusting white balance is shifting the white point along the blue/orange-yellow axes. Some forms of artificial lighting, namely flourescent lighting and CFL bulbs, produce light by running an electric current through a gas. Such lighting often introduces a slight magenta or green tint, and these tints are corrected by shifting the white point along the green/magenta axes.
Both adjustments, tuning RGB curves and moving white point, affect both white balance and color balance, however each affects one much more than the other. A landscape picture style, for example, might introduce a bit more of a green shift in white balance, while enhancing blues and greens from a color balance perspective much more. Correcting the slight green in white balance due to choosing a landscape picture style will affect the overall color balance of the photo as well, however the impact will usually be minor, and the final results are usually still acceptable.
Regarding the subtlety of the changes affected by picture style, thats to be expected, as the differences are fairly subtle. Usually, taking a picture of one single scene with different picture styles will only produce minor differences. On the other hand, taking a portrait with a portrait picture style, and a landscape photo with the landscape picture style, will usually produce better results for those types of photos. The intent of a picture style is to enhance the important color elements of a certain type of photo. Using landscape on a portrait photo probably won't produce anything particularly intriguing beyond a neutral or standard style, while using a portrait style should produce noticeable, even if its only slightly noticeable, improvement in skin tones.