How come at the new site it is $30 to join and at the old site it is $50 to join. I thought they were one in the same?

If you subscribe on the old Caissa, the cost is $50 which includes a one-year premium membership on the old Caissa and a full membership on the new Caissa. You should choose this if you want to play on both the old and new Caissa.
 
If you subscribe on the new Caissa, the cost is $30 for a one year full membership on the new Caissa only. It does not include a membership on the old Caissa.

Does this mean that those who paid for a mere standard membership on the old site aren't members on the new one?
 
Moreover, why is the front page of the new site titled "Caissa's Web _Free_ Online Chess"?

Both standard and premium members on the old Caissa now have an automatic membership on the new Caissa.
 
The "free" refers to the free trial membership, but we may also implement a free membership that does not have a time limit but is very limited in the features it can access.