"Tomorrow" is said in three different ways in Standard Japanese. In the order of formality, those are:
みょうにち、あす and あした. (In kanji, all three are written as 「明日」.)
What that inevitably means is that the native speakers learn the three "words" in the reverse order. 「あした」 is definitely the most intuitive for us.
あした is by far the most common pronunciation for everyday speech among friends, family, neighbors, etc.
あす is a little more formal than あした. It is used in more official communication than daily conversations. Its use in weather forecast on TV and radio, as OP mentioned, is a good example. When adults phone each other to announce an unexpected community or school event for the next day, it is likely that they will use あす over あした. When kids do the same among themselves, however, they will still use あした.
みょうにち is almost reserved for the business world and the most formal occasions. Using あした in business will NOT make you sound like a good businessman. あす would be OK once in a while, but not always in business.
As for the meaning, all three can mean "tomorrow", but only あした and あす can be used to figuratively mean "the future".