Is there any difference between the two? I think さて is more formal than さ/さあ, but is that true? And are there any other nuances?
From a dictionary
As an interjection:
- When moving on to next action, used for talking to himself/herself or invite others. E.g. さて、そろそろ出かけようか
The other two senses listed are rare, so I omitted.
As a conjunction:
- To change the topic E.g., さて、話は変わって
- To move on to next action. Similar to then
- To contradict the preceding sentence. Similar to but
- To invite or urge others. E.g., さあ、始めよう
- To answer vaguely in the negative E.g., さあ、よくわかりません
- Before the new action. E.g., さあ、やるぞ
- When interrupting the conversation partner.
- Same as さあ(1)
- When one is unsure. E.g., さ、これは困った. Similar to さあ(2)
- Same as さあ(4)
Speaking of difference in nuance, さて(1) and さあ(1,3)/さ(1) are relevant. As the definitions suggest
- さて sounds more like the speaker saying to himself, thinking of the next action to take. There is no implied listener.
- さあ sound more like urging. Usually it is assumed that it is directed at someone.
- As such さて sounds less energetic and さあ more energetic.
A concrete example (of inviting さて/さあ) :
- さて出かけようか is just casually calling attention of someone you are supposed to go out; A possible translation is Well, shall we go? If saying to himself, Now it should be time to go.
- さあ出かけようか is similar, but more passionately inviting and expresses your looking forward to going out. Now let's go! It can be used in a saying-to-himself situation to encourage himself, but the use is rarer.
So the difference is not really formality, but さて does give the impression that the speaker is a mature person or at least calm. Probably that is what made you think さて more formal.