This question can not be answered so broadly. I request you to specify a thermodynamic calculation so that it can be answered properly. However, just to point out that "d" or small delta indicates a small change in a parameter whereas Capital Delta would mean a large change usually.

Usually, d is the full differential (infinitely small change) of some parameter, delta is its finite change, small delta can desribe the infinitely small variation of the some parameter, partial derivative shows the change of the value of some thermodynamic function at changing of one its parameter when this function depends on some parameters.

Interesting question. There is no exact definition of what constitutes a large increment or a small decrement. Captial delta is used for larger differences. Small delta is typically used in older books of 50s and 60s to show differences. However, small d and curly d (of partial derivatives, also called Jacobi's delta) have specific meanings. Small d represents the difference when it approaches zero (see definition of limits). The partial derivatives indicate that a property or a quantity is dependent on several variables. If the rest are kept constant and only one variable changes, then curly d is used.

You can read the history of other uses of delta in chemistry in this paper. I asked a similar delta question from a historian 9 years ago!

www.che.uc.edu/jensen/w.%20b.%20jensen/reprints/158.%20Delta.pdf