Perl 5 -pl, 53 bytes

$_.=$".<>;y,\\,/,;s!^(.*)(/.*?) \1/!"../"x$2=~y,/,,!e

Try it online!

JavaScript (Node.js), 88 bytes

s=>d=>s.split(x=/\/|\\/).map(a=>a==d[0]?d.shift():d.unshift`..`,d=d.split(x))&&d.join`/`

Try it online!

from edc65's and fixed some problem

JavaScript (Node.js), 98 bytes

f=(x,y,[[X,a,b],[Y,c,d]]=[x,y].map(e=>/(.*?)[\/\\](.*)|/.exec(e)))=>x?a==c?f(b,d):'../'+f(b,y):[y]

Try it online!

Extra / at last when output is ../../

CJam, 46 bytes

ll]{'\/'/f/:~}/W{)__3$=4$@==}g@,1$-"../"*o>'/*

Try it online.

Examples

$ echo '/usr/share/geany/colorschemes
> /usr/share/vim/vim73/ftplugin' | cjam path.cjam; echo
../../vim/vim73/ftplugin
$ echo 'C:\Windows\System32\drivers
> C:\Windows\System32\WindowsPowerShell\v1.0' | cjam path.cjam; echo
../WindowsPowerShell/v1.0

How it works

ll]         " Read two lines from STDIN and wrap them in an array.                       ";
{           " For each line:                                                             ";
  '\/       " Split by “\”.                                                              ";
  '/f/      " Split each chunk by “/”.                                                   ";
  :~        " Flatten the array of chunks.                                               ";
}/          "                                                                            ";
W           " Push -1 (accumulator).                                                     ";
{           "                                                                            ";
  )__       " Increment and duplicate twice.                                             ";
  3$=       " Extract the corresponding chunk from the first line.                       ";
  4$@=      " Extract the corresponding chunk from the second line.                      ";
  =         " If the are equal,                                                          ";
}g          " repeat the loop.                                                           ";
@,          " Rotate the array of chunks of the first line on top and get its length.    ";
1$-         " Subtract the value of the accumulator.                                     ";
"../"*o     " Print the string “../” repeated that many times.                           ";
>           " Remove all chunks with index less than the accumulator of the second line. ";
'/*         " Join the chunks with “/”.                                                  ";

Bash, 69 66

I didn't post this one because I thought someone must be able to do it much better. But apparently it is not that easy.

sed -r 'N;s/(.*[/\])(.*)\n\1/\2\n/'|sed '1s/[^/\]*/../g;N;s!\n!/!'

N makes sed match two lines together. The first expression removes the common prefix ending with / or \. The second expression replaces directory names with .. in the first line. Finally it concatenates the two lines with the separator.

Thanks to Hasturkun for 3 characters.

Ruby - 89

r=/\/|\\/
s = ARGV[0].split r
d = ARGV[1].split r
puts ("../"*(s-d).size)+((d-s).join"/")

Usage:

ruby relative.rb working/directory destination/directory

Groovy - 144 chars

One solution:

x=args[0][1]!=':'?'/':'\\'
f={args[it].tokenize x}
s=f 0
d=f 1
n=0
while(s[n]==d[n++]);
u="..$x"*(s.size-(--n))
println "$u${d.drop(n).join x}"

example output:

bash$ groovy P.groovy C:\\Windows\\System32\\drivers C:\\Windows\\System32\\WindowsPowerShell\\v1.0
..\WindowsPowerShell\v1.0

bash$ groovy P.groovy /usr/share/geany/colorschemes /usr/share/vim/vim73/ftplugin
../../vim/vim73/ftplugin

bash$ groovy P.groovy /foo/bar/foo/bar /foo/qux/foo/bar
../../../qux/foo/bar

ungolfed:

// fs = file seperator, / or \
fs = args[0][1]!=':'?'/':'\\'

s = args[0].tokenize fs
d = args[1].tokenize fs

// n = # of matching dirs from root + 1
n = 0
while (s[n] == d[n++]) ;

// up = the up-prefix. e.g. "../../..", for instance 
n--
up = "..${fs}" * (s.size-n)

println "$up${d.drop(n).join fs}"

J - 63 char

A function taking the old path on the left and the new path on the right.

}.@;@(c=.c&}.`(,~(<'/..')"0)@.(~:&{.))&('/'<;.1@,'\/'&charsub)~

This solution comes in three parts, looking like post@loop&pre~. Explained by explosion:

post @ loop & pre ~   NB. the full golf
                  ~   NB. swap the arguments: new on left, old on right
            & pre     NB. apply pre to each argument
       loop           NB. run the recursive loop on both
post @                NB. apply post to the final result

'/'<;.1@,'\/'&charsub  NB. pre
         '\/'&charsub  NB. replace every \ char with /
'/'     ,              NB. prepend a / char
   <;.1@               NB. split string on the first char (/)

c=.c&}.`(,~(<'/..')"0)@.(~:&{.)  NB. loop
                      @.(~:&{.)  NB. if the top folders match:
    &}.                          NB.   chop off the top folders
   c                             NB.   recurse
       `                         NB. else:
           (<'/..')"0            NB.   change remaining old folders to /..
         ,~                      NB.   append to front of remaining new folders
c=.                              NB. call this loop c to recurse later

}.@;  NB. post
   ;  NB. turn the list of folders into a string
}.@   NB. chop off the / in the front

Note that we add a leading / to each path before splitting, so that we handle Windows-style paths by making C: into a "folder". This results in an empty folder at the start of Unix-style paths, but that always gets removed by the loop.

See it in action:

   NB. you can use it without a name if you want, we will for brevity
   relpath =. }.@;@(c=.c&}.`(,~(<'/..')"0)@.(~:&{.))&('/'<;.1@,'\/'&charsub)~
   '/usr/share/geany/colorschemes' relpath '/usr/share/vim/vim73/ftplugin'
../../vim/vim73/ftplugin
   'C:\Windows\System32\drivers' relpath 'C:\Windows\System32\WindowsPowerShell\v1.0'
../WindowsPowerShell/v1.0

You can also try it yourself at tryj.tk.

Python 3, 120

a,b=(i.split('\\/'['/'in i])for i in map(input,'  '))
while[]<a[:1]==b[:1]:del a[0],b[0]
print('../'*len(a)+'/'.join(b))

Example:

$ python3 path.py
 /usr/share/geany/colorschemes
/usr/share/vim/vim73/ftplugin 
../../vim/vim73/ftplugin

C, 119 106

void p(char*s,char* d){while(*s==*d){s++;d++;}s--;while(*s){if(*s==47||*s==92)printf("../");s++;}puts(d);}

Ruby >= 1.9, 89 94 characters

$;=/\\|\//
a,b=$*.map &:split
puts"../"*(a.size-r=a.index{a[$.+=1]!=b[$.]}+1)+b[r..-1]*?/

Input via command line arguments. Works for both UNIX- and Windows-style paths, including paths with repeated folder names:

$ ruby relpath.rb /usr/share/geany/colorschemes /usr/share/vim/vim73/ftplugin
../../vim/vim73/ftplugin
$ ruby relpath.rb 'C:\Windows\System32\drivers' 'C:\Windows\System32\WindowsPowerShell\v1.0'
../WindowsPowerShell/v1.0
$ ruby relpath.rb /foo/bar/foo/bar /foo/qux/foo/bar
../../../qux/foo/bar

Javascript (E6) 104

Edit Added alert for output

R=(s,d)=>alert(s.split(x=/\/|\\/).map(a=>a==d[0]?d.shift()&&'':'../',d=d.split(x)).join('')+d.join('/'))

Ungolfed

R (s,d) => // a single espression is returned, no {} or () needed
  s.split(x=/\/|\\/) // split string at / or \, save regexp in X for later
  .map( // create a new array from s
     a => a == d[0] // check if current of s and d equals
          ? d.shift() && '' // map to '' and cut 1 element of d
          : '../', // else map to '../'
     d=d.split(x)) // second param of map is useless, so split d here
  .join('')+d.join('/') // join map and concat to rest of d adding separators

Test

R('C:\\Windows\\System32\\drivers','C:\\Windows\\System32\\WindowsPowerShell\\v1.0')

../WindowsPowerShell/v1.0

R('/usr/share/geany/colorschemes','/usr/share/vim/vim73/ftplugin')

../../vim/vim73/ftplugin

PHP, 158 151

function r($a,$b){$a=explode('/',$a);$b=explode('/',$b);foreach($a as $k=>$v){if($v==$b[$k])$b[$k]='..';else break;}unset($b[0]);echo implode('/',$b);}

Ungolfed:

function r($a,$b){
    $a=explode('/',$a);
    $b=explode('/',$b);
    foreach($a as $k=>$v){
        if($v==$b[$k])$b[$k]='..';
        else break; 
    }
    unset($b[0]);
    echo implode('/',$b);
}
// these lines are not included in count:
r('/test/test2/abc','/test/test3/abcd'); // ../test3/abcd
r('/test/test2/abc','/test/test2/abcd'); // ../../abcd

JavaScript - 155

function p(s,d){s=s.split(/\\|\//g);d=d.split(/\\|\//g);o=[];for(i=0;s[i]==d[i];i++);for(j=s.length-i;j--;)o[j]="..";return o.concat(d.slice(i)).join("/")}

Parses either path format but outputs with / separator.

console.log(p("/usr/share/geany/colorschemes","/usr/share/vim/vim73/ftplugin"));
../../vim/vim73/ftplugin
console.log(p("/usr/share/geany/colorschemes/test/this","/usr/share/vim/vim73/ftplugin/this/is/a/test"));
../../../../vim/vim73/ftplugin/this/is/a/test
console.log(p("c:\\windows\\system32\\drivers\\etc\\host","c:\\windows\\system\\drivers\\etc\host"));
../../../../system/drivers/etchost

Bash + coreutils, 116

Here's a shell script to get the ball rolling. Pretty sure there'll be shorter answers:

n=`cmp <(echo $1) <(echo $2)|grep -Po "\d+(?=,)"`
printf -vs %`grep -o /<<<${1:n-1}|wc -l`s
echo ${s// /../}${2:n-1}

Output:

$ ./rel.sh /usr/share/geany/colorschemes /usr/share/vim/vim73/ftplugin
../vim/vim73/ftplugin
$ ./rel.sh /usr/share/geany/colorschemes/ /usr/share/vim/vim73/ftplugin/
../../vim/vim73/ftplugin/
$ ./rel.sh /usr/share/vim/vim73/ftplugin /usr/share/geany/colorschemes
../../geany/colorschemes
$ 

Note there is no way for the script to tell if the string ftplugin is a file or a directory. You may explicitly provide a directory by appending it with an / as in the example above.

Won't handle paths containing whitespace or other funny characters. Not sure if that is a requirement or not. Just a few extra quotes would be needed.