We study the Piton de la Fournaise (PdF) volcano dynamics through the observation of continuous seismic velocity changes from 2000 to 2013. We compute the cross correlations of ambient seismic noise recorded at more than 30 short-period and broadband stations of the UnderVolc temporary seismic experiment and of the PdF volcano observatory network. The velocity changes are estimated from the travel time delay measured on the cross correlations computed between pairs of stations. We average the relative velocity changes for all pairs of stations and obtain a time series of the velocity change of Piton de la Fournaise volcano over 13 years. From the period 0.5 to 4 s, the depth sensitivity of the velocity change is ranging from approximately 100 m to 2500 m. A slow decrease of velocity is measured from 2000 and ends with a major eruption that occurred in April 2007. This eruptive episode is followed by an increase of the velocity. These long-term changes are compared to the deformation of the Piton de la Fournaise edifice estimated from geodetic measurements. An analysis of baseline change between GPS stations indicates an inflation of the volcanic edifice prior to April 2007 followed by a deflation since then. This deflation predominantly affects the terminal cone. Seismic velocity changes and deformation have similar long-term trends with velocity decrease observed during inflation and velocity increase during deflation. However, the velocity change magnitude is about 2 orders of magnitude greater than the deformation. This suggests nonlinear relation between velocity changes and deformation.