We present here the performance of a simultaneous dual-species matter-wave accelerometer for measuring the differential acceleration between two different atomic species ( 87 Rb and 85 Rb). We study the expression and the extraction of the differential phase from the interferometer output. The differential accelerometer reaches a short-term sensitivity of 1.23 × 10 −7 g/ √ Hz limited by the detection noise and a resolution of 2 × 10 −9 g after 11000 s, the highest reported thus far with a dual-species atom interferometer to our knowledge. Thanks to the simultaneous measurement, such resolution levels can still be achieved even with vibration levels up to 3 × 10 −3 g, corresponding to a common-mode vibration noise rejection ratio of 94 dB (rejection factor of 50 000). These results prove the ability of such atom sensors for realizing a quantum based test of the weak equivalence principle (WEP) at a level of η ∼ 10 −9 even with high vibration levels and a compact sensor.