I think this is a poorly chosen data set for exploring the advantages of zero inflated models, because, as you note, there isn't that much zero inflation.
plot(fitted(fm_pois), fitted(fm_zinb))
shows that the predicted values are almost identical.
In data sets with more zero-inflation, the ZI models give different (and usually better fitting) results than Poisson.
Another way to compare the fit of the models is to compare the size of residuals:
boxplot(abs(resid(fm_pois) - resid(fm_zinb)))
shows that, even here, the residuals from the Poisson are smaller than those from the ZINB. If you have some idea of a magnitude of the residual that is really problematic, you can see what proportion of the residuals in each model are above that. E.g. if being off by more than 1 was unacceptable
sum(abs(resid(fm_pois) > 1))
sum(abs(resid(fm_zinb) > 1))
shows the latter is a bit better - 20 fewer large residuals.
Then the question is whether the added complexity of the models is worth it to you.
I suspect that your problem may be that the default behavior of predict.glm isn't what you think it is.
Specifically, predict used on a glm object will by default gives a response on the scale of the linear predictors, not the response.
This is quite clearly stated in the help (?predict.glm) but seems to trip people up very often (suggesting the default ought to be changed, perhaps; you might like to raise it on the relevant mailing list).
To get the values you want, try predict(model1,type="response")
Best Answer
From another post (here: Error "system is computationally singular" when running a glm), and based on the number of predictor variables you have in your model, I would suggest you look for collinear predictors in your model, or whether you are trying to fit a model with more variables than observations.