Which computing concept did Ada Lovelace describe that Babbage's own notes did not?

That a machine manipulating symbols need not be manipulating numbers. Translating an Italian account of Babbage's Analytical Engine, she added notes three times longer than the original, and in them observed that if the relations between pitched sounds could be expressed in symbols, the engine could compose music - and by extension work on anything representable, not just arithmetic.

Babbage thought of it as a calculating machine. That is the substance of the claim that she saw further: not that she wrote the first program, which is the usual formulation and is arguable, but that she identified the difference between a calculator and a computer.

Note G contains a method for computing Bernoulli numbers, laid out as a table of operations with what amounts to a loop, and is generally called the first published algorithm intended for a machine. How much of it was hers is disputed - Babbage had drafted programs earlier, and their correspondence shows him supplying material and her catching an error of his. She also warned in the same notes that the engine could originate nothing and could only do what we know how to order it to perform, which is quoted against enthusiasts to this day.