Why do results in org mode from look different than what I get on a command line?

The way to see the output of the R process on the script is to specify :results output, instead of the default :results value:

#+begin_src R :results output drawer
month_no <- 1:12
attr(month_no, "names") <- month.name
month_no
#+end_src

#+RESULTS:
:results:
  January  February     March     April       May      June      July    August 
        1         2         3         4         5         6         7         8 
September   October  November  December 
        9        10        11        12 
:end:

With :results output, the evaluation of the block consists of running an R command (in my case (Linux), the command is R --slave --no-save: it's the value of the variable org-babel-R-command which can be customized if necessary), and collecting the standard output of the process.

With :results value the processing is more complicated: 16 lines of code vs. 1 line for the output case.

See the function org-babel-R-evaluate-external-process: in the output case, the processing just runs the R interpreter on the code of the block and gathers up the standard output as the return value (a string); in the value case, it creates some R code on the fly (see the value of org-babel-R-write-object-command) which wraps the block in an R function and calls that function. It then formats the value that is returned and writes it into a temp file. That file is munged by org-babel-R-process-value-result and turns it into a table in Lisp format (in the case above, that's a list of 12 singleton lists: ((1) (2) (3) ... (12)) ) which is then turned into whatever format you specify in the header args (by default, that's an Org table).

These gory details are more-or-less uniform across Babel languages, but there are corner cases which might rear their ugly heads and suprise you. The only way to deal with them is to read the language-specific Babel documentation (see Language Support in Worg - the R documentation can be found here). You should also read the commentary at the top of the ob-<language>.el file - for some languages, but not R, that's the only documentation. For the R case, the commentary can be found here. And eventually, you should consult the code (as I did for this answer).

You should also experiment assiduously: write small blocks, change header arguments and run the examples; ultimately, it's the only sure-fire way to find out what happens.