von-Neumann-Morgenstern v. Bernoulli Utility Function A great deal of time is spent distinguishing the big $U$ (von-Neumann-Morgenstern)v. small $u$ (Bernoulli Utility Function). The v.NM function maps from the space of lotteries to real number as it represents the preference defined on the lottery space while the Bernoulli is defined over sure amounts of money.

Why is this distinction so important in the theory of expected utility? Also, what does this distinction enable us to achieve that is important in the expected utility theory? What is the most intuitive way to understand the expected utility theory is constructed in this way?
Bernoulli utility represents preference over monetary outcomes. In a way, this is no different from the typical utility functions defined over consumption bundles. 

vNM utility, in contrast, represents preference over lotteries of monetary outcomes. Thus, the argument of vNM utility is an object related to, but categorically distinct from, the object that is an argument of Bernoulli utility.  

For example, Bernoulli utility function allows us to compare the utilities from having $\$5$ to having $\$7$, while vNM utility allows us to compare utilities from the lottery $(0.2\otimes\$5,0.8\otimes\$7)$ --- having $\$5$ with $20\%$ and $\$7$ with $80\%$ --- to the lottery $(0.6\otimes\$5,0.4\otimes\$7)$ --- having $\$5$ with $60\%$ and $\$7$ with $40\%$. 

In this sense, the distinction between Bernoulli and vNM utility functions are necessary (as they are applied to different objects) rather than important (since they both represent some kind of preference in the end), as @denesp says in his comment.

Moreover, if we only consider degenerate lotteries, i.e. the probabilities are either $0$ or $1$, then vNM and Bernoulli utilities coincide. 
That is, the Bernoulli utility of having $\$5$ is exactly the same as the vNM utility of having $\$5$ with $100\%$. Therefore, distinguishing Bernoulli from vNM utility functions enables us to examine the effects of uncertainty apart from the mere quantity of "stuff" (be it goods or money). 

Lastly, defining utility over money also allows us to study people's attitudes towards risk.