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One is digital: using numbers, represented by their digits (the adding machine and graph paper she's using). This is how we ordinarily represent numbers, and how we compute with them (e.g., elementary addition, using least significant digits first, then next-most-significant, etc.).

The other is analog: using a slide rule. Here, numbers are represented by a physical magnitude; in this case, length. So rather than π being "3.14159", it is a length of 3.141 units on the slide rule. Analog computing amounts to manipulating these magnitudes, rather than individual digits.

Both methods have been used for millennia, and both have been mechanized. Computing machines were once primarily analog, but now they're almost exclusively digital; we just call those machines computers now. Importantly, they're both real types of computation, although it's easy to forget.