Do you think that using a faster Arduino, as the Due, can make the application work better?
It depends. I suspect in general you will be limited by network speeds on the web server, the speed of writing data to/from one Arduino to the other, and the speed that the SD card can write on the logger. If those are your bottlenecks, then putting a faster processor in won't help at all.
It may be the Due/Mega might be useful in that they would have more memory, and you could do both the logging and the web stuff in one processor. However, that probably requires you to be real careful so that each thread (web serving, writing to the SD card) doesn't interfere with the other.