Confusion regarding wav to MFCC conversion
There are many variations in the ways MFCCs can be extracted. The most common ones are:


STFT parameters like frame rate, FFT size, FFT window.
Filter bank parameters like number of bands, whether filter normalization is used, and whether the filters are applied to the magnitude or power spectra.
Subtle variants in the definition of the discrete cosine transform.


And it is actually not correct to say that there are 13 MFCC - it is indeed very common to keep the 13 first coefficients, but you could very well use a larger cosine transform matrix and get more coefficients. The farther you go, the less information the coefficients will capture (since they will correspond to dimensions of decreasing variance - assuming the discrete cosine transform is a good approximation of a PCA on the mel coefficients data, which it is).

Check this to learn more about all the subtle differences between several "famous" MFCC extraction toolboxes.

In the end, such or such combination of MFCC extraction parameters might give you a difference of at most one or two percentage points at whatever task you are using the MFCC for (for example speech recognition), but there are more interesting things to try to tweak and improve than these parameters.