Basically, can you do everything with the equivalent linear univariate
regression model that you could with the multivariate model?
I believe the answer is no.
If your goal is simply either to estimate the effects (parameters in $\mathbf{B}$) or to further make predictions based on the model, then yes it does not matter to adopt which model formulation between the two.
However, to make statistical inferences especially to perform the classical significance testing, the multivariate formulation seems practically irreplaceable. More specifically let me use the typical data analysis in psychology as an example. The data from $n$ subjects are expressed as
$$ \underset{n \times t}{\mathbf{Y}} = \underset{n \times k}{\mathbf{X}} \hspace{2mm}\underset{k \times t}{\mathbf{B}} + \underset{n \times t}{\mathbf{R}},
$$
where the $k-1$ between-subjects explanatory variables (factor or/and quantitative covariates) are coded as the columns in $\mathbf{X}$ while the $t$ repeated-measures (or within-subject) factor levels are represented as simultaneous variables or the columns in $\mathbf{Y}$.
With the above formulation, any general linear hypothesis can be easily expressed as
$$\mathbf{L} \mathbf{B} \mathbf{M} = \mathbf{C},$$
where $\mathbf{L}$ is composed of the weights among the between-subjects explanatory variables while $\mathbf{L}$ contains the weights among levels of the repeated-measures factors, and $\mathbf{C}$ is a constant matrix, usually $\mathbf{0}$.
The beauty of the multivariate system lies in its separation between the two types of variables, between- and within-subject. It is this separation that allows for the easy formulation for three types of significance testing under the multivariate framework: the classical multivariate testing, repeated-measures multivariate testing, and repeated-measures univariate testing. Furthermore, Mauchly testing for sphericity violation and the corresponding correction methods (Greenhouse-Geisser and Huynh-Feldt) also become natural for univariate testing in the multivariate system. This is exactly how the statistical packages implemented those tests such as car in R, GLM in IBM SPSS Statistics, and REPEATED statement in
PROC GLM of SAS.
I'm not so sure whether the formulation matters in Bayesian data analysis, but I doubt the above testing capability could be formulated and implemented under the univariate platform.
Gelman, Andrew, and Jennifer Hill. Data analysis using regression and multilevel/hierarchical models. Cambridge University Press, 2007, is not about GLMs per se, but also covers that and has a nice mix of theory, hands-on-advice, implementation in R, and exercises (and, when you websearch for it, you might find an ebook version of it!).
Not a textbook, but freely available is this graduate statistics course from the Harvard Government Department, which also covers the most common GLMs. The section videos cover implementation in R. The textbook is King, Gary. Unifying political methodology: The likelihood theory of statistical inference. University of Michigan Press, 1989.
Best Answer
If you need to choose "only one" book I would go with Applied Linear Statistical Models by Neter, Kutner, Nachtsheim and Wasserman. I never owned it (always someone else in the office did) but it never failed to provide me with the some of the best reference material on the multiple regression.
It started from
Simple Linear Regressionproceeded toMultiple Linear RegressionandNon-linear regressionto move ontoSingleand thenMultiple Factor Analysis Studiesto finish off withSpecialized Design Studies(eg. Latin Squares, Response Surface Methodology, etc.). Take notice this is not a small book; at ~1400 page is one of the largest Stats book I have seen.(I have noticed the "Applied Linear Regression Models" suggested earlier. I have not seen that book I suspect the cover a lot of common ground so +1 to that!)
As a second choice a "golden oldie" that was (somewhat) recently rewritten is: Linear Regression Analysis by Seber and Lee. It is a bit theoretical and less applied than the Neter et al. book. If more of a theory is your thing it will worth your time. Do not worry it is not full of asymptotics; just it is less "practical" than the Neter et al. book. :)
I liked the book by Seber more because I found it more concise and easier to use if you already have some basis on linear regression. I accept that it might be less friendly to a newcomer though. Given you have already done a course on simple linear regression you will not have a problem I believe (your Linear Algebra is OK right?).