Wetter

Der momentane (Zeit) Zustand der Atmosphäre (genauer: Troposphäre, die Wetterschicht) in einem bestimmten Ort/einer bestimmten Region.

The state of the atmosphere (troposphäre, the layer responible for the weather) at a specific point in time in a specific region.

Or in bullet points:

weather happens in the atmosphere
at a specific time (for example April 23rd 2018 in the morning)
in a specific area or region for example a city (a whole country would not be specific enough)

Weather prediction is important for many different groups of people:

people who drive cars: it's important for them to know if there is fog or if there is snow or ice on the road.
pilots: they need to know about storms to avoid turbulences.
ship captains: they also need to know about storms or fog to prepare themselves, their cargo or their passengers.
farmers: they need to know about frost or hail to protect their plants from damage.

There are weather stations to get all the required data for weather prediction.

Wheater stations are:

In Austria are 250 weather stations. So you need lots of stations to get enough weather data.

Climate

Unlike weather climate isn't about a temperature at a specific time in a specific place. Climate is about calculating averages for example:

The average amount of rain for Vienna (place: a single city) for the year 2012 (time: a whole year).
The average number of sunny days in Burgenland (place: a wider region) for June 2016 (time: a whole month).
The average temperature for Austria (place: a whole country) for the last 10 years (time: a long time).

temperature - thermometer (°C)
pressure - barometer (millibar, short: mbar)
amount of rainfall - rain gauge in German Regenmesser (mm, yes millimeter)
wind speed and wind direction - windsock
Windstärke und Windrichtung - Windsack (Himmelsrichtungen, Stärke nicht präzise genug.)
humidity (Luftfeuchtigkeit)- hygrometer