寒い and 冷たい: cold surroundings or a cold object
Distinguish 寒い for cold weather and surroundings from 冷たい for the temperature of a specific object or a cold manner. Both can appear in the same scene because they describe different perspectives.
寒い: feel cold in an environment
寒い is used when low temperature makes the weather or a surrounding space feel cold. 今日は風が強くて寒いです。 and この部屋は暖房がなくて寒いです。 focus on the weather and the room as environments.
A speaker can say コートを着てもまだ寒いです。 when still feeling cold despite wearing a coat. This differs from describing the surface temperature of one specific thing such as a hand or some water.
冷たい: cold to the touch
冷たい commonly describes the low temperature of something sensed or touched, such as water, a hand, or metal. 井戸の水は夏でも冷たいです。 and 雪に触ったので手が冷たいです。 make the water and the hand the cold objects.
It can also describe an unkind or emotionally distant response. 彼の冷たい返事に妹は黙りました。 refers to the cold impression of the reply, not to the physical temperature of a voice.
Separate the perspectives in one scene
外は寒いですが、川の水はもっと冷たいです。 uses 寒い for the outdoor weather and 冷たい for the temperature of the water. English may use cold for both, but Japanese marks what is being described.
冷たい風 foregrounds how the wind feels against the skin, while 寒い日 describes a day with cold weather. Actual usage does not always form an absolute boundary, so identify what is cold and who experiences the surrounding chill.
Quick check: After touching snow, which is more natural for saying that your hand is cold, 手が寒いです or 手が冷たいです?
Show the answer and why
手が冷たいです is natural because it describes the low temperature of a specific body part. Use 寒いです when the speaker feels cold in the surrounding environment.