Data gathered on a rating or interval scale is considered the most precise, easy to compare and analyse. The rating scale of 1-10 allows you to do more advanced analysis on the results, like comparing the mean and mode. You can better test and prove correlations and generally use the data in more ways depending on how deep you want to go.
Using a scale of 1-10 will allow you to get more accurate, nuanced responses. You will notice less migration to the middle range as is often witnessed on scales because 1-10 has no true middle and the wider 1-10 scale vs. the 1-5 scale means there is more variation and subtle change measured over time. Given The Happiness Index is a pulse measurement tool; the wider 1-10 scale made this the optimal range.