In Debian, there are some compiled R packages in the official repositories. But one could also install a R package from source. I am interested to know why would a user prefer one method of installation to another.
3 Answers
It's sometimes preferable to 'compile' the sources on your server rather than just using an existing executable file.
This is because the compiler makes the exe file specifically for your machine so may run faster and work much better, for instance the compiler knows the processor you have so can optimise for this.

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The idea of pre-compiled R packages for Debian/Ubuntu is borrowing from Windows and MacOS. Those OSes have pre-compiled packages since they typically don't have the standard tools in standard locations for building packages from source (c and fortran compilers, latex, perl, etc.).
If there is a new release of a package on CRAN, is the pre-compiled package on Debian repos automatically updated? I believe that you better sync with CRAN. Check out the package ctv
to help you manage large collections of R packages ("CRAN views"), both for installing and updating.
You need root privileges to install a pre-compiled package from the OS repos, while any regular user may install any packages using install.packages()
in R (but I recommend to run sudo R, if you are the sysadmin, for installing CRAN views, so as to make them available system-wide, instead of inflating your ~/).
One inconvenient to source packages is that if you fetch many, the compiling will take extra time to install (depending on your machine). You might gain in performance from compiling, but it is not guaranteed to be noticeable.

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I already provided a somewhat detailed answer in response to this SO question.
As an update, these days you even have lots of packages prebuilt thanks to updated cran2deb initiaives:
On Ubuntu you now have almost all CRAN packages prebuilt via Michael Rutter's 'cran2deb for ubuntu' ppa on Launchpad.
For Debian, Don Armstrong now provides a similar service (also covering BioConductor and OmegaHat) at debian-r.debian.net.

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13This answer do not say anything about "Difference between installing a package from source and from compiled binary" – Sambit Tripathy Mar 24 '15 at 17:08