Will the received waves be the same if the transmitter use the beamforming to transmit the same signal?
Say you have the receive antennas very close together, like with spacing $\lambda/10$ or similar and they are uniformly spaced in a linear array. As you pointed out the the phase shifts between them will be fairly small, so what does it mean? It means that the spatial resolution of your beamformer will be poor. In the extreme case, consider something crazy where the spacing is $\lambda/1000$, all the receive antennas essentially receive the same copy of the signal. So why can't you just add them all up if they have no phase difference? You can, you just won't have any spatial discrimination and it is only really amplifying your signal by the number of receive elements. 

I will leave a plot here that I generated in MATLAB which is the take away. You see as the spacing (specified in wavelengths) becomes smaller and smaller the beam response becomes flatter and flatter indicating that the receive array is unable to spatially discriminate because of the reasons outlined earlier. Hope this helps!