fft function in R vs spec function from 'seewave' package? They don't give a similar frequency spectra
Disclaimer: I don't know even a bit of R. But I checked the documentation of the spec function: A Hanning function is applied to the analysis window. 

This may very well be the source of the difference in the two plots. The fft function does not apply any window (i.e., it's only the natural rectangular window we are seeing).