空 is commonly used and means sky. 天 roughly refers to the space above the sky. c.f. 天国（てんごく） or sometimes 天 itself means heaven. However, 天気=weather and others are just words. 大空 means big sky, not being hindered by clouds, mountains and so on. P.S. We never say ありがとう=thank you (in advance) at the end. You can say よろしくおねがいします instead, which sort of means thank you in this case.

空 is just sky. 天 is also sky but more like mystery. Like we call our emperor as 天皇. King is 王様 or 王. As you know emperor is higher hierarchy than king. if you put 空 to both of them, 大空 is just sky, but 天空 is more like mystery. One of the famous Gibli movie, Castle in the Sky. Its Japanese name is 天空の城ラピュタ. Castle in the sky what a mysterious! If Japanese title is 大空の城ラピュタ. Sounds boring lol

天: abstract 空: concrete