Why should you use Paris traceroute?
Because traceroute fails in the presence of routers that employ load balancing on packet header fields. The failures lead to the discovery of inaccurate and incomplete paths, that may mislead operators during problem diagnosis and result in erroneous internet maps.
Paris traceroute, by controling packet header contents, obtains a more precise picture of the actual routes that packets follow.
For a quick demonstration, look at the figure on the right. Suppose you are trying to measure the route between Src and Dst. The true router topology is shown on the left. L is a router that balances load on two paths, via routers A or C. The middle of the figure shows what you might see with classic traceroute. The right part is what you would get with Paris traceroute...

A brief demonstration of Paris traceroute's skills