Pure plane-wave basis sets have the following advantages when used in periodic DFT (or HF) simulations:

• Orthogonal
• Computationally simple (operators with derivatives are particularly straightforward)
• Low-scaling methods allow easy transformations between real- and reciprocal-space
• Basis set size does not scale with electron count
• Geometry-independent
• Their accuracy is controlled with a single parameter, and it is systematically improvable
• Model all space with equal accuracy

However there are some disadvantages:

• Basis set size scales with simulation volume - vacuum is not "free"
• Basis sets are usually large "per atom" - it is not usually practical to construct the full Hamiltonian explicitly (or any other operator) and you must solve the eigenequations iteratively
• Model all space with equal accuracy - no scope for focusing effort on "interesting" regions
• Extend throughout space (no simple real-space truncation possible in integrals - e.g. the Fock operator is computationally expensive)

In contrast, (periodic) local basis sets generally have the following advantages:

• Basis set size does not scale with simulation volume
• Basis set is typically compact, with few basis states "per atom"
• Model space with variable accuracy - basis can be tuned to improve representation in regions of interest, and reduce accuracy in uninteresting regions
• Basis functions are local, and real-space truncations are straightforward in multi-basis set integrals
• Some basis choices (e.g. Gaussians) allow analytic integration of some energy terms

and the following disadvantages:

• Non-orthogonal
• Computationally complicated (often)
• They are geometry-dependent (leading to Pulay forces)
• Basis set size scales with electron count
• Model space with variable accuracy - need to decide a priori where to spend the computational effort, i.e. which regions are "interesting"
• No single parameter to control their accuracy; not always systematically improvable
• Some basis set choices are not easy to transform between real- and reciprocal-space

Roughly speaking, plane-wave methods are efficient when computing and applying the terms of the Hamiltonian, but lead to a much larger dimensionality in the eigenvalue problem and must compute a subset of states; local basis sets often take more time constructing the eigenvalue problem, but it is quite compact and can be solved directly (e.g. with LAPACK) to generate the full eigenspectrum.

There is no reason in principle why you cannot use a hybrid approach (e.g. like CP2K) whereby you transform to a different basis set to perform certain parts of the calculation. You can gain some of the advantages of both, but unfortunately you may suffer from some of the disadvantages of both as well -- for example, when switching from plane-waves to Gaussians the Fock operator becomes much more compact and computationally tractable, but you need to ensure that there are Gaussians in all the "interesting" regions of space. The computational cost of the transformation can also be problematic.

Two final comments:

• "Muffin tin" programs use mixed basis sets, using localised basis functions to represent the regions of space near nuclei, and plane-waves in the interstitial regions. This is efficient in both regions, but matching the descriptions at the boundaries can be tricky

• Wannier transformations allow a "lossless" transformation of the occupied Kohn-Sham states from a plane-wave representation to a local representation. However, the transformation scales cubically and is not well-conditioned, usually relying on a "guess" transformation which would be generated from a local basis set (typically LCAO)

Pure plane-wave basis sets have the following advantages when used in periodic DFT (or HF) simulations:

• Orthogonal
• Computationally simple (operators with derivatives are particularly straightforward)
• Low-scaling methods allow easy transformations between real- and reciprocal-space
• Basis set size does not scale with electron count
• Independent of atomic positions
• Their accuracy is controlled with a single parameter, and it is systematically improvable
• Model all space with equal accuracy

However there are some disadvantages:

• Basis set size scales with simulation volume - vacuum is not "free"
• Basis sets are usually large "per atom" - it is not usually practical to construct the full Hamiltonian explicitly (or any other operator) and you must solve the eigenequations iteratively
• Model all space with equal accuracy - no scope for focusing effort on "interesting" regions
• Extend throughout space (no simple real-space truncation possible in integrals - e.g. the Fock operator is computationally expensive)

In contrast, (periodic) local basis sets generally have the following advantages:

• Basis set size does not scale with simulation volume
• Basis set is typically compact, with few basis states "per atom"
• Model space with variable accuracy - basis can be tuned to improve representation in regions of interest, and reduce accuracy in uninteresting regions
• Basis functions are local, and real-space truncations are straightforward in multi-basis set integrals
• Some basis choices (e.g. Gaussians) allow analytic integration of some energy terms

and the following disadvantages:

• Non-orthogonal
• Computationally complicated (often)
• They depend on the atomic positions (leading to Pulay forces)
• Basis set size scales with electron count
• Model space with variable accuracy - need to decide a priori where to spend the computational effort, i.e. which regions are "interesting"
• No single parameter to control their accuracy; not always systematically improvable
• Some basis set choices are not easy to transform between real- and reciprocal-space

Roughly speaking, plane-wave methods are efficient when computing and applying the terms of the Hamiltonian, but lead to a much larger dimensionality in the eigenvalue problem and must compute a subset of states; local basis sets often take more time constructing the eigenvalue problem, but it is quite compact and can be solved directly (e.g. with LAPACK) to generate the full eigenspectrum.

There is no reason in principle why you cannot use a hybrid approach (e.g. like CP2K) whereby you transform to a different basis set to perform certain parts of the calculation. You can gain some of the advantages of both, but unfortunately you may suffer from some of the disadvantages of both as well -- for example, when switching from plane-waves to Gaussians the Fock operator becomes much more compact and computationally tractable, but you need to ensure that there are Gaussians in all the "interesting" regions of space. The computational cost of the transformation can also be problematic.

Two final comments:

• "Muffin tin" programs use mixed basis sets, using localised basis functions to represent the regions of space near nuclei, and plane-waves in the interstitial regions. This is efficient in both regions, but matching the descriptions at the boundaries can be tricky

• Wannier transformations allow a "lossless" transformation of the occupied Kohn-Sham states from a plane-wave representation to a local representation. However, the transformation scales cubically and is not well-conditioned, usually relying on a "guess" transformation which would be generated from a local basis set (typically LCAO)