Is there a difference between these two words for "tomorrow" and when each is used? (and is it just coincidence that あした sounds like the past tense of あす?)
We tend to be taught あした and then discover あす when we listen to the programs like the weather forecast. I've never seen an explanation. My dictionary cross references them to each other.
"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".
I'm going to translate this best answer on goo
First, a bit on the origins: あすdescribes the "next day", ie the day after today. On the other handあしたdescribed the "morning" in contrast to nightfall. Since the sun setting or nightfall signified the end of the day, あした was used to refer to the morning after. Now-a-days, we use the word 翌朝（よくあさ）to describe the next morning. Eventually, instead of referring to just "morning", あした became to describe tomorrow (AM) and finally became to mean tomorrow. Today, あした is used casually and in the order of あした→あす→みょうにちfor formality.
In answering your question: Although they mean the same thing, in terms of usage, あした is most often used colloquially while あす is commonly used in writing. The kanji 明日 is read as あす in most dictionaries and is said to be read that way to emphasize how it is written.
