What is the difference between water and organic solvents?

Like dissolves like. Polar solvents dissolve polar solutes best, and non-polar solvents dissolve non-polar solutes best. Water is a polar molecule, and it therefore dissolves other polar molecules or ionic compounds. Organic solvents are generally not as polar as water or completely non-polar, so things like hydrocarbons, high molar mass alcohols and molecular compounds dissolve best in them.

Also, water can hydrogen bond extremely well, which results in some of its unique properties, such as it's high boiling point (especially for it molar mass) and it's high surface tension. Typically organic solvents have much lower boiling points and a lower surface tension.

It's also worth mentioning that water is completely non-toxic and environmentally benign, whereas most organic solvents are fairly toxic, especially if ingested, and many are carcinogenic.