First-order Greeks answer what happens if one input moves a little. Second-order Greeks answer what happens to those answers. They matter because every hedge is built from first-order numbers that are only true for an instant.
Retail traders can safely ignore most of them and still do well, provided they respect gamma and understand that delta and vega are moving targets. Desks cannot ignore them, because a large book's profit and loss is dominated by exactly these terms.
Example: you model a short XYZ strangle's overnight risk using only delta and vega and conclude a 3% gap costs $4,000. Include gamma, vanna and the volatility jump that accompanies such a gap and the real figure is closer to $9,000. The difference is entirely second order.