What is the exact difference between the $\hat{u}$ in PRF vs SRF? I've been confused about the exact difference between PRF and SRF is. 

I know that it has to do with other variables that influence the regression that is not accounted for, but what is the difference in the models?



 
PRF: Population Regression Function
SRF: Sample Regression Function.
Population Regression Function (PRF) is the relation between conditional expectation of population $Y_{i}$ conditional on population $X_{i}$, versus the corresponding $X_{i}$, which is also a random variable (to be precise, a matrix of random variables) with some specific given values, say $x$, that is, $E(Y_{i}|X_{i}=x)= f(X_{i})$, which is PRF. For linear regression, $f(X_{i})=\beta_{0}+\beta_{1}X_{i}$.

For a specific value $X_{i}=x$ in population, there could be multiple corresponding $Y_{i}$. (Similarly, imagine when we have a sample dataset and we sort by $X$, we can find multiple corresponding $Y$). Thus, using the relation we get above, we have: $Y_{i}=E(Y_{i}|X_{i}=x)+\mu_{i}$ (Note: it's $\mu$, the deviation of population $Y_{i}$ from its conditional expectation value, rather than $\widehat{\mu }$, which is an estimated value from a specific sample). In other words, PRF captures the perfect, or ideal situation where if $X=x$ happens, we get $Y=y$. But remember both $Y_{i}$ and $X_{i}$  are random variables that may deviate from the perfect situation. Given $X_{i}=x$, the model uses an error term $\mu_{i}$ to capture the deviation caused by other random factors other than $X_{i}$.

Sample Regression Function (SRF), on the other hand, is the estimated relation between estimated $Y_{i}$, denoted as $\widehat{Y_{i}}$, meaning it is calculated by SRF, that is, $\widehat f(X_{i})=\widehat\beta_{0}+\widehat\beta_{1}X_{i}$, and $X_{i}$. (Note that $X_{i}$ and $Y_{i}$ here are actual values from sample, rather than from population, although we usually use the same denotations). $\widehat\beta_{0}$ and $\widehat\beta_{1}$ mean they are calculated or estimated from sample, rather than the actual relation between population $Y_{i}$ and $X_{i}$. Also, $\widehat f(X_{i})$, $\widehat\beta_{0}$ and $\widehat\beta_{1}$ are estimated parameters/relation based on sample, in order to "guess" the true parameters/relation in population. The estimated error term $\widehat{\mu}$, is the difference between the sample $Y_{i}$ and the calculated value $\widehat{Y_{i}}$ using the model we specify, and is the estimator of population ${\mu}$.