We extract gravitational waveforms from numerical simulations of black hole binaries computed using the Spectral Einstein Code. We compare two extraction methods: direct construction of the Newman-Penrose (NP) scalar $Psi_4$ at a finite distance from the source and Cauchy-characteristic extraction (CCE). The direct NP approach is simpler than CCE, but NP waveforms can be contaminated by near-zone effects---unless the waves are extracted at several distances from the source and extrapolated to infinity. Even then, the resulting waveforms can in principle be contaminated by gauge effects. In contrast, CCE directly provides, by construction, gauge-invariant waveforms at future null infinity. We verify the gauge invariance of CCE by running the same physical simulation using two different gauge conditions. We find that these two gauge conditions produce the same CCE waveforms but show differences in extrapolated-$Psi_4$ waveforms. We examine data from several different binary configurations and measure the dominant sources of error in the extrapolated-$Psi_4$ and CCE waveforms. In some cases, we find that NP waveforms extrapolated to infinity agree with the corresponding CCE waveforms to within the estimated error bars. However, we find that in other cases extrapolated and CCE waveforms disagree, most notably for $m=0$ memory modes.
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