SML, 176 bytes

open OS.FileSys;val! =chDir;fun&n w=(print("\n"^w^n);!n;print"/";c(openDir(getDir()))(w^"\t");!"..")and c$w=case readDir$of SOME i=>(&i w handle _=>();c$w)|x=>()fun%p=(&p"";!p)

Declares (amongst others) a function % which takes a string as argument. Call with % "C:/Some/Path"; or % (getDir()); for the current directory.

I'm using the normally rather functionally used language StandardML whose FileSys-Library I discovered after reading this challenge.

The special characters !, &, $ and % have no special meaning in the language itself and are simply used identifiers; however the can't be mixed with the standard alphanumeric identifiers which allows to get rid of quite some otherwise needed spaces.

open OS.FileSys;
val ! = chDir;                       define ! as short cut for chDir

fun & n w = (                        & is the function name
                                     n is the current file or directory name
                                     w is a string containing the tabs
    print ("\n"^w^n);                ^ concatenates strings
    ! n;                             change in the directory, this throws an 
                                     exception if n is a file name
    print "/";                       if we are here, n is a directory so print a /
    c (openDir(getDir())) (w^"\t");  call c with new directory and add a tab to w
                                     to print the contents of the directory n
    ! ".."                           we're finished with n so go up again
)
and c $ w =                          'and' instead of 'fun' must be used 
                                     because '&' and 'c' are mutual recursive
                                     $ is a stream of the directory content
    case readDir $ of                case distinction whether any files are left
        SOME i => (                  yes, i is the file or directory name
            & i w handle _ => ();    call & to print i an check whether it's a 
                                     directory or not, handle the thrown exception 
            c $ w )                  recursively call c to check for more files in $
        | x    => ()                 no more files, we are finished

fun % p = (                          % is the function name, 
                                     p is a string containing the path
    & p "";                          call & to print the directory specified by p
                                     and recursively it's sub-directories
    ! p                              change back to path p due to the ! ".." in &
)

Can be compiled like this with SML/NJ or with Moscow ML* by prefixing with load"OS";.

*See mosml.org, can't post more than 2 links.