Peter's significance to Clarissa is complicated. On the one hand, he is significant as her past love, the passion forever relinquished. On the other hand, during the time-frame of the story, his significance is to help confirm her place and choice in life: she has a talent and ability and it is valuable in its own right, even if she doesn't always contemplate "Wagner and Pope."
She did think it mattered, her party [...] The curtain with its flight of birds of Paradise blew out again. And Clarissa saw--she saw Ralph Lyon beat it back, and go on talking. So it wasn’t a failure after all! [...] they went on, they went into the rooms; into something now, not nothing,
In a greater sense, Peter is significant because he is the embodiment of Clarissa's life conflict of self against a powerful external influence, that influence being Peter himself.
Clarissa says Peter was always thinking of what was wrong with Clarissa's soul and disagreeing with her about what was important in life: "the defects of her own soul. How he scolded her! How they argued!" They argued so about it that, on one day Clarissa reminisces about, Peter made her cry alone in her bedroom: "she had cried over it in her bedroom."
Yet Peter charms her and has an emotional influence over her: "he could be intolerable; ... but adorable to walk with on a morning like this." Her challenge as a young woman was to exert her self or to sacrifice her self to Peter's influence and yield to his power. Clarissa won the conflict: "she had to break with him or they would have been destroyed." She rejected Peter and chose to marry Richard who gave her independence ("in marriage a little licence, a little independence there must be") and respect.
Richard allowed Clarissa to exercise her gits for what they were as she knew best how to use them: "[she] couldn’t help feeling that she had, anyhow, made this [party] happen." Clarissa's self won the conflict against Peter's influence. What was lost while self was gained was gaiety.
all in a clap it came over her, If I had married him, this gaiety would have been mine all day!