It would be the different reactions they had to the change in thinking. For example First generation poets Coleridge and Wordsworth tended to privilege emotion over rationality etc, whilst second generation composers such as Austen began to present intense passion and emotion as resulting in destruction. In a way warning the audience of the dangerous nature of the Romantic thinking. Sorry if this doesn't help. Try talking to your teacher
Gravedigging! Check the dates first
I'll just say though, OP asked for difference between 1st/2nd generation
poets. So we have for first gen, Coleridge/Wordsworth/Blake (Blake perhaps somewhat the odd one out) and for second gen, Shelley/Byron/Keats. I don't particularly agree with what you have to say about the second generation, since Shelley as an idealist often expressed passionate and surging emotions.
I would say the first generation directly challenged Neoclassical ideas, instead advocating a retreat to nature whilst also promoting the power of the imagination and individual emotion. In simple words, they took the ordinary in life and made it extraordinary (e.g. through the notion of the sublime, etc.). Second generation poets instead criticised the flaws of the first generation (not the flaws/nature of Romantic thinking - 'Romanticism' isn't defined by some preordained list of ideals, it's defined by the composers, the texts and the ideals that arose throughout the entire era). They weren't as 'disillusioned' with the French Revolution as perhaps the first generation was, after seeing the destructive consequences, but kept the 'spirit' of the rebellion and desire for change. This somewhat links with the view of the role of the poet as a "prophetic visionary" (versus 1st generation concerns with the everyday).
Keep in mind this is generalised, as each composer has their own unique characteristics and slight differences.
I would also argue against Austen being a second generation composer. I don't think she should be classified a 'Romantic', since she still retained many values of Neoclassicism. Her work was simply written during the Romanticism movement. It's also difficult to define anyway which generation she would belong to, since she began writing when the first generation had established themselves, yet was still writing when the second generation emerged. She died before many of the initial significant works of the second generation though, so I would be more hesitant to associate her with them.