What is the difference between a semi-permeable and a partially permeable membrane?

There is no real difference other than the wording; both semi-permeable and partially-permeable mean the ability to allow some substances to pass.

However, OCR examiners for AS biology have said that you should "not use the term semi-permeable when referring to cell membranes that allow water and other solutes to cross them" but should "refer to them as partially permeable."