ATLAS measurements of the production of muons from heavy-flavor decays in $\sqrt{{s}_{NN}}=2.76$ TeV Pb+Pb collisions and $\sqrt{s}=2.76$ TeV $pp$ collisions at the LHC are presented. Integrated luminosities of 0.14 ${\mathrm{nb}}^{\ensuremath{-}1}$ and 570 ${\mathrm{nb}}^{\ensuremath{-}1}$ are used for the Pb+Pb and $pp$ measurements, respectively, which are performed over the muon transverse momentum range $4<{p}_{T}<14$ GeV and for five Pb+Pb centrality intervals. Backgrounds arising from in-flight pion and kaon decays, hadronic showers, and misreconstructed muons are statistically removed using a template-fitting procedure. The heavy-flavor muon differential cross sections and per-event yields are measured in $pp$ and Pb+Pb collisions, respectively. The nuclear modification factor ${R}_{AA}$ obtained from these is observed to be independent of ${p}_{T}$, within uncertainties, and to be less than unity, which indicates suppressed production of heavy-flavor muons in Pb+Pb collisions. For the 10% most central Pb+Pb events, the measured ${R}_{AA}$ is approximately 0.35. The azimuthal modulation of the heavy-flavor muon yields is also measured and the associated Fourier coefficients ${v}_{n}$ for $n=2$, 3, and 4 are given as a function of ${p}_{T}$ and centrality. They vary slowly with ${p}_{T}$ and show a systematic variation with centrality which is characteristic of other anisotropy measurements, such as that observed for inclusive hadrons. The measured ${R}_{AA}$ and ${v}_{n}$ values are also compared with theoretical calculations.