The basic difference between Standard and Vivid screen mode can be deducted from each screen mode name. Vivid screen mode simply provides more intensive colors.
Standard screen mode - is also known as Natural screen mode. It provides the most accurate colors, what gives a natural look to images and videos.
Vivid screen mode - is an enhanced color mode, which is displaying more vibrant and saturated colors. It makes colors richer and highly visible even in direct sunlight. Vivid mode improves the sharpness, contrast and brightness.
It is difficult to exactly say if natural or vivid screen mode is better. It depends on user's preferences. Natural screen mode is considered to be more comfortable for eyes, when watching the screen for a longer period of time.