What is the difference between zero-inflated and hurdle models? I wonder if there is a clear-cut difference between the so-called zero-inflated distributions (models) and so-called hurdle-at-zero distributions (models)? The terms occur quite often in the literature and I suspect they are not the same, but would you please explain me the difference in simple terms?  
 A: Hurdle models assume that there is only one process by which a zero can be produced, while zero-inflated models assume that there are 2 different processes that can produce a zero.
Hurdle models assume 2 types of subjects: (1) those who never experience the outcome and (2) those who always experience the outcome at least once.  Zero-inflated models conceptualize subjects as (1) those who never experience the outcome and (2) those who can experience the outcome but don't always.
In simple terms: both zero-inflated and hurdle models are described in two parts.
The first is the on-off part, which is a binary process.  The system is "off" with probability $\pi$ and "on" with probability $1-\pi$.  (Here, $\pi$ is known as the inflation probability.)  When the system is "off," only zero counts are possible.  This part is the same for zero-inflated and hurdle models.
The second part is the counting part, which occurs when the system is "on."  This is where zero-inflated and hurdle models differ.  In zero-inflated models, counts can still be zero.  In hurdle models they must be nonzero.  For this part, zero-inflated models use a "usual" discrete probability distribution while hurdle models use a zero-truncated discrete probability distribution function.
Example of a hurdle model: An automobile manufacturer wants to compare two quality control programs for its automobiles.  It will compare them on the basis of the number of warranty claims filed.  For each program, a set of randomly selected customers are followed for 1 year and the number of warranty claims they file is counted.  The inflation probabilities for each of the two programs are then compared.  The “off” state is “filed zero claims” while the “on” state is “filed at least one claim.”
Example of a zero-inflated model:  In the same study above, the researchers find out that some repairs on the automobiles were fixed without the filing of a warranty claim.  In this way, the zeroes are a mixture of the absence of quality control problems as well as the presence of quality control problems that involved no warranty claims.  The “off” state means “filed zero claims” while the “on” state means “filed at least one claim OR had repairs fixed without filing a claim.”
See here for a study in which both types of models were applied to the same data set.
A: Regarding hurdle models, here's a quote from Advances in Mathematical and Statistical Modeling (Arnold, Balakrishnan, Sarabia, & Mínguez, 2008):

The hurdle model is characterized by the process below the hurdle and the one above. Obviously, the most widely used hurdle model is the one that sets the hurdle at zero. Formally, the hurdle-at-zero model is expressed as: $P(N_i=n_i)=f_1(0)$ for $n_i=0$ $P(N_i=n_i)=\frac{1-f_1(0)}{1-f_2(0)}f_2(n_i)=\phi f_2(n_i)$ for $n_i=1,2,...$
The variable $\phi$ can be interpreted as the probability of crossing the hurdle, or more precisely in the case of insurance, the probability to report at least one claim.

As for zero-inflated models, Wikipedia says:

A zero-inflated model is a statistical model based on a zero-inflated probability distribution, i.e. a distribution that allows for frequent zero-valued observations.
The zero-inflated Poisson model concerns a random event containing excess zero-count data in unit time.$^{[1]}$ For example, the number of claims to an insurance company by any given covered person is almost always zero, otherwise substantial losses would cause the insurance company to go bankrupt. The zero-inflated Poisson (ZIP) model employs two components that correspond to two zero generating processes. The first process is governed by a binary distribution that generates structural zeros. The second process is governed by a Poisson distribution that generates counts, some of which may be zero. The two model components are described as follows: $$\Pr (y_j = 0) = \pi + (1 - \pi) e^{-\lambda}$$
$$\Pr (y_j = h_i) = (1 - \pi) \frac{\lambda^{h_i} e^{-\lambda}} {h_i!},\qquad h_i \ge 1$$
where the outcome variable $y_j$ has any non-negative integer value, $\lambda_i$ is the expected Poisson count for the $i$th individual; $\pi$ is the probability of extra zeros.

From Arnold and colleagues (2008), I see that a hurdle-at-zero model is a special case of the more general class of hurdle models, but from a reference on Wikipedia (Hall, 2004), I also see that some zero-inflated models can be upper-bounded. I don't quite understand the difference in the formulas, but they seem to be quite similar (both even use a very similar example, insurance claims). I hope other answers can help explain any important difference(s), and that this answer will help set the stage for those.
Wikipedia's reference:

*

*Lambert, D. (1992). Zero-inflated Poisson regression, with an application to defects in manufacturing. Technometrics, 34(1), 1–14.

A: in ZIP model $y_i$~0 with probability $\pi$ and $y_i$~ Poisson($\lambda$) distribution with probability $1-\pi$, 
thus the ZIP model is mixture model with 2 components and:
$$\Pr (y_j = 0) = \pi + (1 - \pi) e^{-\lambda}$$
$$\Pr (y_j = x_i) = (1 - \pi) \frac{\lambda^{x_i} e^{-\lambda}} {x_i!},\qquad x_i \ge 1$$
and in an hurdle model $y_i$~ 0 with probability $\pi$ and $y_i$~ truncated Poisson($\lambda$) distribution with probability $1-\pi$, and: 
$$\Pr (y_j = 0) = \pi $$
$$\Pr (y_j = x_i) = \frac{(1 - \pi)} {1-e^{-\lambda}} (\frac{\lambda^{x_i} e^{-\lambda}} {x_i!}),\qquad x_i \ge 1$$
A: Thank you for the interesting question!
Difference: One  limitation  of standard  count models  is  that  the zeros and  the nonzeros (positives) are assumed  to  come from  the same data-generating process.  With hurdle models, these two processes are not constrained  to  be  the  same. The basic  idea is  that  a Bernoulli probability  governs the binary outcome  of  whether  a  count  variate  has a  zero  or positive realization.  If the  realization  is positive, the  hurdle is  crossed,  and  the  conditional  distribution  of the positives is governed  by a  truncated-at-zero  count  data  model. With zero-inflated models, the response variable is modelled as a mixture of a Bernoulli distribution (or call it a point mass at zero) and a Poisson distribution (or any other count distribution supported on
non-negative integers). For more detail and formulae, see, for example, Gurmu and Trivedi (2011) and Dalrymple, Hudson, and Ford (2003).
Example: Hurdle models can be motivated by sequential decision-making processes confronted by individuals. You first decide if you need to buy something, and then you decide on the quantity of that something (which must be positive). When you are allowed to (or can potentially) buy nothing after your decision to buy something is an example of a situation where zero-inflated model is appropriate. Zeros may come from two sources: a) no decision to buy; b) wanted to buy but ended up buying nothing (e.g. out of stock).
Beta: The hurdle model is a special case of the two-part model described in Chapter
16 of Frees (2011). There, we will see that for two-part models, the amount of health care utilized may be a continuous as well as a count variable. So what has been somewhat confusingly termed "zero-inflated beta distribution" in the literature is in fact belongs in the class of two-part distributions and models (so common in actuarial science), which is consistent with the above definition of a hurdle model. This excellent book discussed zero-inflated models in section 12.4.1 and hurdle models in section 12.4.2, with formulas and examples from actuarial applications.
History: zero-inflated Poisson (ZIP) models without covariates have a long history (see e.g., Johnson and Kotz, 1969). The general form of ZIP regression models incorporating covariates is due to Lambert (1992). Hurdle models were first proposed by a Canadian statistician Cragg (1971), and later developped further by Mullahy (1986). You may also consider Croston (1972), where positive geometric counts are used together with Bernoulli process to describe an integer-valued process dominated by zeros.
R: Finally, if you use R, there is package pscl for "Classes and Methods for R developed in the Political Science Computational Laboratory" by Simon Jackman, containing hurdle() and zeroinfl() functions by Achim Zeileis.
The following references have been consulted to produce the above: 


*

*Gurmu, S. & Trivedi, P. K. Excess Zeros in Count Models for Recreational Trips Journal of Business & Economic Statistics, 1996, 14, 469-477

*Johnson, N., Kotz, S., Distributions in Statistics: Discrete Distributions. 1969, Houghton MiZin, Boston

*Lambert, D., Zero-inflated Poisson regression with an application to defects in manufacturing. Technometrics, 1992, 34 (1), 1–14.

*Cragg, J. G. Some Statistical Models for Limited Dependent Variables with Application to the Demand for Durable Goods Econometrica, 1971, 39, 829-844

*Mullahy, J. Specification and testing of some modified count data models Journal of Econometrics, 1986, 33, 341-365

*Frees, E. W. Regression Modeling with Actuarial and Financial Applications Cambridge University Press, 2011

*Dalrymple, M. L.; Hudson, I. L. & Ford, R. P. K. Finite Mixture, Zero-inflated Poisson and Hurdle models with application to SIDS Computational Statistics & Data Analysis, 2003, 41, 491-504

*Croston, J. D. Forecasting and Stock Control for Intermittent Demands Operational Research Quarterly, 1972, 23, 289-303

