If you would like to use grep to find all of the occurences of $pat just like the m//, then split the line before passing to grep, as shown below:

#!/usr/bin/perl -w
use strict;

my $count1;
my $count2;
my $pat = "apple";
my @apples;
while (<DATA>) {
        @apples = grep /$pat/, split ' ', $_;
        $count1 += @apples;
        @apples = m/$pat/gi;
        $count2 += @apples;
}
print "Count one is $count1\n";
print "Count two is $count2\n";

__DATA__
awna apple apple glais
16 Uriore Rd apple
Highapplegate corner, VT. 4xyz59
[download]

Prints:

Count one is 4
Count two is 4
[download]