Actually, it requires very close to twice as much memory. Try it for yourself:

## Create a test datafile of ~ 4MB
perl -E"say 'x 'x10 for 1 .. 2e5" > junk.dat

## Then load it into an array of arrays using a while loop 
## and check the memory consumed using the Task Manager or TOP 
perl -E"$n=0;$a[$n++]=[split] while $_=<>; <STDIN>" junk.dat
## On my system the process has used 214.8 MB

## Now do the same thing using map
perl -E"@a=map[split],<>; <STDIN>" junk.dat
## On my system this process has used 345.1 MB.
[download]

With map,

1. First one list of all the lines in the file is contructed;
2. From that a second (output) list of all the references to small arrays is contructed;
3. Finally, that list is assigned to the array.

Whilst the final AoA will consume the same amount of memory in both cases, the memory consumed by the intermediate lists will have considerably increased thhe overall memory required to construct the final array.

And depending upon your OS, the time taken to build it can be considerable longer using map.

Sorry, I should've been more precise.  Perl constructs an intermediate list of arrays, which is held on Perl's stack (in its entirety) before it is assigned to the final array. This temporarily consumes a lot of memory, which is not returned to the OS (at least not with typical Unix-builds of Perl). Of course, the memory is returned to Perl's own memory pool, so it may be reused later. But the peak memory usage of the process increases.

As soon as references are involved, the additional temporary data only involves the first level elements, i.e. the array references in this case. The referenced data isn't being duplicated, of course.  Still, if the ratio of the references to the payload data is bad (i.e. scond-level arrays with only few non-complex items), there can still be considerable overhead.

Just for comparison, for anyone interested:

sub mem {
    print "$_[0]:\n";
    system "/bin/ps", "-osize,vsize", $$;
}

my @a;
my %tests = (
    # immediate data - no references
    iter_flat =>    
        sub {
            my $n = shift;
            push @a, $_*42 for 1..$n;
        },
    func_flat =>
        sub {
            my $n = shift;
            @a = map $_*42, 1..$n;
        },

    # indirect/referenced data
    iter_ref =>
        sub {
            my $n = shift;
            push @a, [ $_*42 ] for 1..$n;
        },
    func_ref =>
        sub {
            my $n = shift;
            @a = map [ $_*42 ], 1..$n;
        }
);

my $what = shift @ARGV;
my $n    = shift @ARGV || 10_000_000;

mem("before");
$tests{$what}->($n);
mem("after");
[download]

$ ./883539.pl iter_flat         
before:
   SZ    VSZ
  608  22032
after:
   SZ    VSZ
395196 416620

$ ./883539.pl func_flat
before:
   SZ    VSZ
  608  22032
after:
   SZ    VSZ
1390892 1412316           #  map needs 3.5 times as much


$ ./883539.pl iter_ref 
before:
   SZ    VSZ
  608  22032
after:
   SZ    VSZ
1547832 1569256

$ ./883539.pl func_ref
before:
   SZ    VSZ
  608  22032
after:
   SZ    VSZ
2571632 2593056           #  map needs 1.7 times as much
[download]