Part of the problem is that there are two different omega coefficients. Omega_h and Omega_t. I just checked semTools and they seem to report omega_t (which they label as omega).
psych reports both. With the example data set from semTools, the answers are similar, but not identical. I suspect the difference is the psych version allows cross loadings and the sem version does not.
reliability(fit)
visual textual speed total
alpha 0.6261171 0.8827069 0.6884550 0.7604886
omega 0.6253180 0.8851754 0.6877600 0.8453351
omega2 0.6253180 0.8851754 0.6877600 0.8453351
omega3 0.6120052 0.8850608 0.6858417 0.8596204
avevar 0.3705589 0.7210163 0.4244883 0.5145874
summary(omega(HolzingerSwineford1939[7:15],covar=TRUE))
Omega
Alpha: 0.76
G.6: 0.81
Omega Hierarchical: 0.44
Omega H asymptotic: 0.52
Omega Total 0.85
...
Total, General and Subset omega for each subset
g F1* F2* F3*
Omega total for total scores and subscales 0.85 0.89 0.65 0.71
Omega general for total scores and subscales 0.44 0.27 0.24 0.17
Omega group for total scores and subscales 0.35 0.62 0.40 0.54
For a discussion of various estimates of reliability see the article on PsyArXiv preprints: reliability from Alpha to Omega preprint of reliability article available from psyArXiv