After the identification of the gamma-ray blazar TXS 0506+056 as the first compelling IceCube neutrino source candidate, we perform a systematic analysis of all high-energy neutrino events satisfying the IceCube realtime trigger criteria. We find one additional known gamma-ray source, the blazar GB6 J1040+0617, in spatial coincidence with a neutrino in this sample. The chance probability of this coincidence is 30% after trial correction. For the first time, we present a systematic study of the gamma-ray flux, spectral and optical variability, and multiwavelength behavior of GB6 J1040+0617 and compare it to TXS 0506+056. We find that TXS 0506+056 shows strong flux variability in the Fermi-Large Area Telescope gamma-ray band, being in an active state around the arrival of IceCube-170922A, but in a low state during the archival IceCube neutrino flare in 2014/15. In both cases the spectral shape is statistically compatible (≤2σ) with the average spectrum showing no indication of a significant relative increase of a high-energy component. While the association of GB6 J1040+0617 with the neutrino is consistent with background expectations, the source appears to be a plausible neutrino source candidate based on its energetics and multiwavelength features, namely a bright optical flare and modestly increased gamma-ray activity. Finding one or two neutrinos originating from gamma-ray blazars in the given sample of high-energy neutrinos is consistent with previously derived limits of neutrino emission from gamma-ray blazars, indicating the sources of the majority of cosmic high-energy neutrinos remain unknown.