Ha! That's an open-ended question! You ask about "differences" - from what? In September-October you need to be prepared for almost all weather conditions but I would expect the weather to be mostly warm and sunny. I carry virtually the same things whether it is March, April, May, September, October, November. I haven't walked in the other months, but would expect only minor variations.
To "cross the Pyrenees" or other higher-elevation areas on the Camino, you do not need any special items. You should have a hat or buff anyway, and socks can be used as gloves if you don't have gloves. You will be there in early September, which is late summer. Check in the pilgrim office to confirm that the weather conditions are acceptable. On one occasion at the end of October, I found it very cold and windy but my rain jacket, gloves and rain pants kept me safe.
You should take several layers of lightweight clothes - if you have unusually cold weather on one or 2 days, you can put on all your layers - a couple of shirts, rain jacket with a hood, hat or buff, socks on your hands. You don't need "different" items - just layer them all on.
Be sure to keep one complete set of clothes, including a long-sleeved shirt, dry for the evening. I take a sleeveless down vest as an extra layer for evenings and sleeping, not for walking.