I have a problem assessing my zero-inflated negative binomial model against a poisson model

Question

I have replicated a zero-inflated negative binomial model (model 1, table 1) from the article:

Scharpf, Adam. 2020. “Why Governments Have Their Troops Trained Abroad: Evidence from Latin America.” International Studies Quarterly 64 (3): 734–47. https://doi.org/10.1093/isq/sqaa043

The data can be found here: https://dataverse.harvard.edu/file.xhtml?fileId=5117253&version=1.0

My problem lies in the assessment and comparison of model 1 and a poisson model (with the same data).

I have successfully made a histogram of observed and predicted outcomes of the original model 1 in order to assess the model, but when I make the histogram of predicted outcomes for the poisson model I get the same spread as with the zero-inflated negative binomial. I know something isn't right because I've also assessed the models with rootograms, and there is a clear difference...

I'll upload the code for the different models:

Zero-inflated negative binomial:

library(pscl)
library(dplyr)
library(ggplot2)

datayus <- read.table("https://dataverse.harvard.edu/api/access/datafile/5117253?format=tab&gbrecs=true", header=TRUE, sep="\t")

model1 <- zeroinfl(soaGEN ~
                      lguerrillad +
                      lstrikesd +
                      ldemond +
                      lriotsd +
                      ldmidtoward | lfpsimusa,
                    dist="negbin",
                    datayus)

datayus$pred.model1 <- predict(model1,
                          newdata = datayus,
                          type = "response")

datayus %>% 
  ggplot() +
  geom_histogram(aes(x = soaGEN), alpha = 0.5) +
  geom_histogram(aes(x = pred.model1), fill = "red", alpha = 0.3)+
  theme_minimal() +
  ggtitle("Fordeling mellem observerede og forudsagte outcomes") +
  ylab("Frekvens") +
  xlab("Deltagelse på SOA")

Poisson model:

mod.pois <- glm(soaGEN ~
                  lguerrillad +
                  lstrikesd +
                  ldemond +
                  lriotsd +
                  ldmidtoward + lfpsimusa,
                family ="poisson",
                datayus)

datayus$pred.pois <- predict(mod.pois,
                        newdata = datayus,
                        "response")

datayus %>% 
  ggplot() +
  geom_histogram(aes(x = soaGEN), alpha = 0.5) +
  geom_histogram(aes(x = pred.pois), fill = "red", alpha = 0.3) +
  theme_minimal() +
  ggtitle("Fordeling mellem observerede og forudsagte outcomes") +
  ylab("Antal optrædener") +
  xlab("Deltagelse på SOA")

I hope someone will be able to help.