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We present new isochrone fits to the colour-magnitude diagrams of the Galactic globular clusters NGC 6362 and NGC 6723. We utilize 22 and 26 photometric filters for NGC 6362 and NGC 6723, respectively, from the ultraviolet to mid-infrared using data sets from Hubble Space Telescope, Gaia, unWISE, and other photometric sources. We use models and isochrones from the Dartmouth Stellar Evolution Database (DSED) and Bag of Stellar Tracks and Isochrones (BaSTI) for α-enhanced [α/Fe] = +0.4 and different helium abundances. The metallicities [Fe/H] = -1.04 ± 0.07 and -1.09 ± 0.06 are derived from the red giant branch slopes in our fitting for NGC 6362 and NGC 6723, respectively. They agree with spectroscopic estimates from the literature. We find a differential reddening up to ΔE(B - V) = 0.13 mag in the NGC 6723 field due to the adjacent Corona Australis cloud complex. We derive the following for NGC 6362 and NGC 6723, respectively: distances 7.75 ± 0.03 ± 0.15 (statistic and systematic error) and 8.15 ± 0.04 ± 0.15 kpc; ages 12.0 ± 0.1 ± 0.8 and 12.4 ± 0.1 ± 0.8 Gyr; extinctions A_V = 0.19 ± 0.04 ± 0.06 and 0.24 ± 0.03 ± 0.06 mag; reddenings E(B - V) = 0.056 ± 0.01 ± 0.02 and 0.068 ± 0.01 ± 0.02 mag. DSED provides systematically lower [Fe/H] and higher reddenings than BaSTI. However, the models agree in their relative estimates: NGC 6723 is 0.44 ± 0.04 kpc further, 0.5 ± 0.1 Gyr older, ΔE(B - V) = 0.007 ± 0.002 more reddened, and with 0.05 ± 0.01 dex lower [Fe/H] than NGC 6362. The lower metallicity and greater age of NGC 6723 with respect to NGC 6362 explain their horizontal branch morphology difference. This confirms age as the second parameter for these clusters. We provide lists of the cluster members from the Gaia Data Release 3.