Why do we use ANOVA and linear regression together? What is the advantage (if any) to fitting both a linear regression model to a dataset and running an ANOVA test if these are both ultimately identical? 
 A: Tested hypotheses in ANOVA and linear regression are different.


*

*Linear model gives one estimates of the coefficient values along with
the significance of their being different from zero. In this way one 
can understand how each of the input variables (as factor levels if we refer to a linear model with categorical input variable) influences the mean value of a continuous dependent variable.


So the null hypothesis is being formulated for each input variable as to whether they differ from zero. Additionally you can formulate a similar one-sided hypothesis for R Squared value.


*

*ANOVA's null hypothesis is whether group mean values of the dependent
variable are not significantly different, while an alternative
hypothesis is just that at least one of the factor level forms a
group of observations which mean value is different from overall
mean. ANOVA does not specify which of the factor levels are
significant discriminators.


If ANOVA test results in rejecting the null hypothesis, you can build a significant linear regression model, but before you build one, you don't understand which coefficient(s) will be significantly different from zero.
Besides, Fisher's (F) statistic used in ANOVA applies to measuring a significant decrease in Residual Sum of Squares attributed to the model one has built in comparison with intercept-only model.
