In this paper, we analyse the characteristics of the elliptic instability in a finite cylinder in the presence of both background rotation and axial stratification. A general formula for the linear growth rate of the stationary sinuous modes is derived including viscous and detuning effects in the limit of small eccentricity. This formula is discussed and compared to experimental results which are obtained in a cylinder filled with salted water for two different eccentricities by varying the stratification, the background rotation and the cylinder rotation. A good agreement with the theory concerning the domain of instability of the sinuous modes is demonstrated. Other elliptic instability modes, oscillating at the cylinder angular frequency are also evidenced together with a new type of instability mode, which could be connected to a centrifugal instability occurring during the experimental phase of spin-up. The nonlinear regime of the elliptic instability is also documented. In contrast with the homogeneous case, no cycle involving growth, breakdown and re-laminarization is observed in the presence of strong stratification. The elliptic instability in a stratified fluid seems to yield either a persistent turbulent state or a weakly nonlinear regime.