The words "actually is a CW-complex" suggest to me that the CW-structure is known, whereas simply having the homotopy type of a CW-complex suggests the CW-structure is unknown, or at least not uniquely determined. So one answer to your question could be, "compute invariants that are defined using the CW-structure".

I have in mind cellular homology and cohomology, of course. But there are other examples from homotopy theory, such as various types of Hopf invariant, boundary maps in cofibre sequences, the Leray-Serre and Atiyah-Hirzebruch spectral sequences, $\ldots$

The words "actually is a CW-complex" suggest to me that the CW-structure is known, whereas simply having the homotopy type of a CW-complex suggests the CW-structure is unknown, or at least not uniquely determined. So one answer to your question could be, "compute invariants that are defined using the CW-structure".

I have in mind cellular homology and cohomology, of course. But there are other examples from homotopy theory, such as various types of Hopf invariant, boundary maps in cofibre sequences, the Leray-Serre and Atiyah-Hirzebruch spectral sequences, $\ldots$

$\ldots$ and Whitehead torsion (see John Klein's answer).

The words "actually is a CW-complex" suggest to me that the CW-structure is known, whereas having the homotopy type of a CW-complex suggests the CW-structure is unknown, or at least not uniquely determined. So one answer to your question could be, "compute invariants that are defined using the CW-structure".

I have in mind cellular homology and cohomology, of course. But there are other examples from homotopy theory, such as various types of Hopf invariant, boundary maps in cofibre sequences, Whitehead products, $\ldots$

The words "actually is a CW-complex" suggest to me that the CW-structure is known, whereas simply having the homotopy type of a CW-complex suggests the CW-structure is unknown, or at least not uniquely determined. So one answer to your question could be, "compute invariants that are defined using the CW-structure".

I have in mind cellular homology and cohomology, of course. But there are other examples from homotopy theory, such as various types of Hopf invariant, boundary maps in cofibre sequences, the Leray-Serre and Atiyah-Hirzebruch spectral sequences, $\ldots$