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.
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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.