Difference between time series prediction vs point process prediction I am working on a problem of predicting event counts based on user history. 
This is a classical time series analysis problem, and I used the ARIMA model: (wiki).
I also applied a Hawkes point process model for the same purpose. Therein I predict all the points using a univariate Hawkes process and calculate counts. 
However, I do not understand the fundamental difference between these two models;
I do know that ARIMA takes previous event counts as input, and the Hawkes model takes event timestamps as input.
Would someone please point out what the other differences between these models are? 
 A: *

*Let's start with AR. AR by default is a time-discrete model, HP or general point process can be defined on continuous space, such as a timeline. In practice, point process data can be discretized using time bins. The data counts in each interval can be either binary or integers.

*In AR, $X_t$ is a linear combination of the past values $X_{t-1}, X_{t-2},...$ with additive Normal noise. In HP, the underlying intensity function is a linear combination of past events. To be more specific, 
$$\lambda_t = \mu+ \alpha_1 X_{t-1} + \alpha_2 X_{t-2} + ... $$
usually people write this as
$$\lambda_t = \mu+ \sum_i h(t - i)X_i  $$
$h(t - i)$ can be seen as $\alpha_i$.
The probability of observing $n$ events in time interval $t$ follows Poisson distribution,
$$ \mathbb{P}( X_t =n ) \sim \text{Poisson}(\lambda_t \Delta) $$
$\Delta$ is the width of the time interval.
In brief, the noise of the observation in HP is not Gaussian.

*Come back to ARIMA, HP does not incorporate error terms in the past (no "MA" part), not to mention the temporal difference (no "I" part). You can add something in HP like the "MA" and "I" components, but I've never seen that.

A: A time series has time measurements made at regular time intervals, whereas in a Poisson process, including the Hawkes process, the time measurements are distributed in a Poissonian way.
