Basically, the Heckit model is supposed to model data that is zero-inflated due to sample bias while the distribution of the population is not zero-inflated, while the zero-inflated negative binomial regression is used if the true distribution of the population is actually zero-inflated. As a conclusion, the Heckit model and the zero-inflated negative binomial regression are not alternatives to each other but suitable for completely different situations.
The following two citations nicely wrap this up:
Theoretically the standard Tobit model is applicable only if the
underlying dependent variable contains negative values that have been
censored to zero in the empirical realization of the variable. In
practice, though, the Tobit modelis routinely employed when the values
of the observed dependent variable are exclusively nonnegative and are
clustered at zero, irrespective of whether any censoring has
occurred. The Heckit model has emerged as the de facto default
alternative to Tobit when values cluster at zero due to selection bias rather than censoring, but applications of the Heckit model have proven problematic as well.
Sigelman, L., & Zeng, L. (1999). Analyzing Censored and Sample-Selected Data with Tobit and Heckit Models. Political Analysis, 8(2), 167-182. doi:10.1093/oxfordjournals.pan.a029811
The zero-inflated negative binomial (ZINB) regression is used for
count data that exhibit overdispersion and excess zeros. The data
distribution combines the negative binomial distribution and the logit
distribution. The possible values of Y are the nonnegative integers:
0, 1, 2, 3, and so on.
From the NCSS Statistical Software (Developers of PASS Sample Size) Chapter 328. Link.