Relative to the (O. Carm.) Carmelites of Ancient Observance, I suggest you start here:
ocarm.org/en/content/ocarm/welcome
For the (OCD) Teresian reform…the Discalced Carmelites…I suggest you start here:
discalcedcarmel.com/quienes-somos/historia/desde-el-monte-carmelo
With regard to both websites, note especially the menus that give the various topic headings that should help to answer your questions.
For your specific questions about the differences…I would recommend this article from the British Province of the Ancient Observance.
carmelite.org/index.php?nuc=content&id=166
Today, especially, there is a great closeness between the two branches of Carmel. Each has its own Prior General in Rome; they are separate and distinct but they work more closely together today and without the tensions that were once there.
Each branch has:
Friars (First Order)
Contemplative nuns (Second Order)
Laity/Permanent Deacons/Diocesan Clergy (Third Order Secular)
Additionally, there are also active Religious (male and female) who operate under either the ancient observance or the discalced observance and secular institutes which are part of the Carmelite family.