When, why and how did the bronc saddles used in rodeo change from
having a saddle horn to being the hornless type used today?

I have photos of amatuer contestants in 1942 using a Hamley
Association saddle with a horn.  Also a photo from 1947 with a
saddle(unknown maker) with a horn.  I know in the 1920's a committee
from several rodeos went to Hamley Saddle company and designed a
regulation saddle (with a horn) that was provided to the contestants
if they did not have their own.  The first hornless saddle was made in
1922 but was not widely used.

From one rodeo fan to another, great question.  Here's the answer from
the National Cowboy & Western Heritage Museum.  The search on the site
is not user-friendly, but it'll do the job if you're patient with it. 
There's a picture of the hornless saddle on the site.

Bronc Riding Saddle 
Hamley Saddlery Co., 1933 
H.130 

Developed in 1919 by the Pendleton, Cheyenne, Walla Walla and Boise
rodeo committees (in consultation with Hamley Saddlery), "Committee"
or "Association" saddles became standard equipment in the bronc riding
event. Built on the old Ellensburg tree, the saddle was made without a
roping horn and incorporated a somewhat undercut, swell-fork pommel
and a slightly dished, five-inch cantle. This specimen belonged to
bronc rider Harry Knight.

http://www.nationalcowboymuseum.org/g_rode_high.html