We have done it both ways. The first time, I booked the first night in advance in SJPDP and nothing else (not even Roncesvalles where we arrived at 8:00 p.m. because we had not anticipated walking in snow in April). Everything worked out great!! We never had a "no vacancy" problem with the exception of having to search a little harder for an empty bed in Palas de Rei, however, that was mid-April to mid- May.
The second time we left SJPDP on May 31, so I was very concerned about the summer crowds I had been hearing about (as well as the commentary made about Peter's snoring by our albergue mates on the previous walk) so I booked everything in advance. It was completely unnecessary until after Sarria. In fact, we stayed an extra day in Sarria and we were able to book a private room at an albergue that morning although by the end of the day it was full (mostly because a large school group was staying there). It seemed to me there were less people walking in June than in May, but that may have been because the weather was alternating between torrential rains and sweltering heat.
There are so many reasons not to book in advance such as wanting to stay an extra day someplace (or not wanting to because there is finally a break in the heat wave or lightening), needing a rest, illness , not knowing what your pace is. . . .I had also pre-booked the entire route for a much younger traveling companion and ended up cancelling all of his reservations after Roncesvalles because he would walk 30 miles and then rest a couple of days which is a pattern I did not anticipate when I was making reservations (he also got sick for three days). Fortunately, most of the changes I had to make to my "best laid plans" were far enough in advance that I did not have to pay for a cancelled night. I used booking.com and I like it - having said that, it is very difficult to change a reservation to the next night when one uses booking.com. It would have been very easy to do had I booked directly with the albergue or hotel.
Finally, I will just add one more difference in our experience of booking ahead (and staying in private rooms) vs. looking for a place to stay as you go. I feel like on our overly planned second walk we missed out on some of the most memorable experiences of our unplanned first walk, like stumbling upon Casa Magica and sharing a great laugh with our roommates even though we didn't understand German and Romanian and they didn't speak English.
Nonetheless, both walks were wonderful and all that pre-planning was just another great lesson learned along the Camino!
I hope that helps! Buen Camino!