The choice is related to you goal from extraction.
If you need to extract thermolabile substances (enzymes, ....), you should use cold water (example: maceration for 24h) to protect your substances from modification. If effect of temperature is neglected regarding your isolated substances, then use hot water (example: heat reflux extraction for 2h) which gives you more yield in short time.
It can be both cold water and hot water, depending on the type of plants and extracts to be obtained (as the mentioned example of the action of the temperature in the participation in the chemical and physicochemical reaction of thermolabile enzymes). Even, it is often necessary to use other solvents than water to obtain the active ingredients, since not all are water-soluble.
Cold extraction is used to extract peptides. In this case, we work to mix at 4 ° C to avoid oxidation . Hot extraction is generally used to extract polar molecules