Why would one number theorems, propositions and lemmas separately?

If the paper contains three main theorems, each generalizing the previous, it is nice to be able to discuss them like this:

While the extension of Theorem 1 to Theorem 2 uses only complex analysis, in Theorem 3 we will have to employ some Ramsey theory.

This is a slight elaboration of François Dorais's comment. If you have a small number of theorems/lemmas/propositions—let's say, small enough that readers can reasonably be expected to hold all the theorems in their head at once—then the second method of numbering can help readers grasp the flow of the paper and can even serve as a mnemonic aid.

A secondary consideration, similar to what Fedor Petrov said, is that the reader may want to skim through and just look at the main theorems. If you adopt the first method of numbering, then readers might accidentally skip from (say) Theorem 8 to Theorem 17 without realizing that they missed Theorem 14.

One famous book that uses the second method of numbering is Serre's Course in Arithmetic. Serre uses the "Theorem" designation very sparsely in that book, and the numbering system helps make the Theorems stand out.