I don't know if it is the grease though.
I always thought that using heavy greases and oils (with a few exceptions) with heat would fry your hair. I know that using petroleum jelly with high heat will fry the hair because it's a fabulous heat conductor.
I think that the difference between now and then was just usage. Firstly, I almost never got my hair pressed. Only on my birthday, school picture day, and other rare special occasions. I feel like people straighten their hair more often now, so they're more prone to heat damage.
And when my mom straightened my hair, she was very careful with how hot she let it get and how long she left it in my hair because it didn't come with any built in safeguards. It was a hunk of metal with a handle. As long as she left it on the stove, it'd get hotter and hotter, so she was very vigilant about watching it. Now, I think that most people aren't so concerned with precautions, because of all the extra bells and whistles on heat styling appliances. All the ionized heat, infrared lights, ceramic/tourmaline plates, heat dials, blah blah blah lulls people into a false sense of security, and I feel like they don't watch what they're doing as carefully. Regular folks and stylists alike have gotten real comfortable using ridiculously excessive amounts of heat because as long as they use their $500 machine (or a $20 machine with "ceramic" plates) to do it, nothing is going to happen.
And as one of the ladies said already, getting hair straight enough to put into stiff ringlets and ponytails takes far less heat than getting hair silky, swangy, Pocahantas straight. Ergo, less potential for heat damage.
But all that aside, looking back on my childhood, I feel like there were lots of little girls running around with heat-trained hair. Nobody identified it as damaged, because damage implies that something is wrong with the hair, and to folks, heat training the hair was a good thing. Hell, people purposefully trained their children's hair. I had classmates who would brag about their trained hair and how "easy" it was. But a difference in perception doesn't change the fact that folks have been losing their texture to heat appliances for a while now. It's not so new.