The followings are what a dictionary says. 天気 is used for weather for a specific time or day or a short period of time like several days, whereas 天候 is used for weather for a period from several days to dozens of days or a season. However, I would use 天気 for about a week as the longest, whereas 天候 for between a week and a season. Here are some examples: ・明日の天気は曇りでしょう。 あすのてんきは　くもりでしょう。 It will be cloudy tomorrow. ・ここ1週間の天気はあまりよくない。 ここいっしゅうかんの　てんきは　あまりよくない。 It hasn't been fine in this week. ・去年の夏の天候は冷夏だった。 きょねんのなつのてんこうは　れいかだった。 It was cold during last summer. ・この3月の天候は穏やかになるらしい。 このにがつ、さんがつの　てんこうは　おだやかに　なるらしい。 It will seem to be mild weather this coming March. 天気 also means a fine weather. ・今日はお天気になってよかったね。 きょうは　おてんきになって　よかったね。 It's good that we have a nice weather today, isn't it? I hope this will be of any help!

we use 天気 instead 天候. 天候 is sometime used in weather news.

天気ーconsidering sunny, cloudy or rain 天候ーconsidering not only sunny, cloudy or rain, even temperature, wind and moisture etc