When to apply time series models? When should I apply traditional time series models (e.g. additive, ARIMA) versus other models such as linear regression? 
For example, if I wanted to build a predictive model for website traffic, I could approach it with:


*

*Use previous traffic data + time series technique like exponential smoothing

*Collect other data features such as advertising spend, social media activity, email campaigns and run a linear regression model
The former approach is more straight forward, faster to implement and likely to fit better compared to the latter. What disadvantages am I not taking into account here? 
The only thing I'm coming up with is that approach #2 will provide some insight into the weighting of the feature data (e.g. relationship between ad spend and traffic).
 A: The answer you are looking for is completely dependent on what you are attempting to model.  
If you are looking to forecast future website traffic without changing anything in your website, then use some sort of time series forecasting algorithm.  This ensures you will not have to worry about annoying autocorrelation or other external factors.  
If you are interested in understanding what factors drive the website traffic, you will want something a bit more in-depth, which allows you to include and interpret external variables.  A simple example is linear regression, which allows you to interpret variance in website traffic by a number of input variables (one of which may be time-of-day).

The former approach is more straight forward, faster to implement and likely to fit better compared to the latter. What disadvantages am I not taking into account here?

This may not always be true, but in some cases it is.  Statistical modeling packages are quite robust and this makes for easy time series modeling.  However, using a simple ARIMA model may not actually predict very well without giving you extremely high errors.  Plus, traditional ARIMA does not allow you to incorporate external factors driving website traffic; it is completely dependent on previous values of your measure of interest.  So, if you edit your website in any way, your forecast could be very very inaccurate (since you are not accounting for external factors which could have a huge impact on traffic). 

The only thing I'm coming up with is that approach #2 will provide some insight into the weighting of the feature data (e.g. relationship between ad spend and traffic).

Understanding how different variables are affecting your output (website traffic) could be very important if you are interested in, say, increasing traffic in the future, not just modeling future traffic.  For example, if one of your variables is percentage_of_blue_on_page, and you find that it is inversely linearly related to the amount of website traffic you are seeing, you may want to try to decrease the amount of blue on your page to increase website traffic.  This is a silly example, of course, but it is meant to be demonstrative.
