I personally feel like $array[] is cleaner to look at, and honestly splitting hairs over milliseconds is pretty irrelevant unless you plan on appending hundreds of thousands of strings to your array.

I ran this code:

$t = microtime(true);
$array = array();
for($i = 0; $i < 10000; $i++) {
    $array[] = $i;
}
print microtime(true) - $t;
print '<br>';
$t = microtime(true);
$array = array();
for($i = 0; $i < 10000; $i++) {
    array_push($array, $i);
}
print microtime(true) - $t;

The first method using $array[] is almost 50% faster than the second one.

Some benchmark results:

Run 1
0.0054171085357666 // array_push
0.0028800964355469 // array[]

Run 2
0.0054559707641602 // array_push
0.002892017364502 // array[]

Run 3
0.0055501461029053 // array_push
0.0028610229492188 // array[]

This shouldn't be surprising, as the PHP manual notes this:

If you use array_push() to add one element to the array it's better to use $array[] = because in that way there is no overhead of calling a function.

The way it is phrased I wouldn't be surprised if array_push is more efficient when adding multiple values. Out of curiosity, I did some further testing, and even for a large amount of additions, individual $array[] calls are faster than one big array_push. Interesting.