You are confusing weatherproof and freezeproof. Some digital cameras are weatherproof but not freezeproof, although all current freezeproof cameras are weatherproof.

Weatherproof generally means that the camera can be splashed with water from any direction without water entering the camera. There are standards to measure this but most camera makers are rather vague when describing their weatherproof cameras. Keep in mind that:

• Not all weatherproof cameras are equally solid. Some only handle a little rain, some plenty and some even can be rinsed with some pressure.
• Weatherpoof is not waterproof either. Unless a camera is waterproof, it cannot be submerged.
• For an interchangeable lens camera, you must use a weatherproof camera and a weatherproof lens for the whole thing to be weatherproof.

For a non-weatherproof camera, there are no guaranties but most will handle snow and even a few drops of rain. You don't have to worry for each drop but you should wide the camera dry as often as possible. If water enters the camera, it can cause a short-circuit and permanently damage it. There are rain-covers you can buy for non-weatherproof cameras to protect them.

Freezeproof means the camera can operate normally below freezing, usually up to -10C (14F). This is the temperature the camera has been tested against but this not mean it will stop operating below. All other cameras are expected to operate normally until 0C (32F). Below the minimum operating temperature:

• Battery life will be significantly shorter and will eventually stop working entirely. This is temporary in most cases. When the battery warms up, it will generate current again. A trick I use it to keep the second battery in my glove so that it stays warm, when the one in the camera stops working, I swap the batteries and repeat this until they are actually depleted.
• A battery which freezes completely can be permanently damaged. Unfortunately you cannot tell by looking at it. From experience (I live in Montreal), down to -20C for a freezeproof camera or to -10C for one that isn't there are no problems.
• The body or lens are unlikely to be damaged by cold unless it is extreme. Some lenses though sometimes become very hard to zoom and stop focusing as the internal lubricant freezes.
• The LCD and EVF are mostly made of liquid crystals which slow down and eventually freeze. They will start being less responsive and eventually not form an image until warmed up again.

There are new OLED displays. I have no idea how those will react, if someone knows, feel free to add details to this answer.