There is no difference. “Purchase” and “buy” are synonyms (they mean the same thing).
Andrea is right, buy and purchase are the same.
However, a better question would be “what is the difference between a lease option and a lease purchase agreement?”
Lease option is you lease a home with the option to buy the property at the end of the lease
Lease purchase is similar to lease option EXCEPT, with a lease purchase, buying is not an option but rather a requirement.
Hope this helps!