("a", "b") and qw(a, b)

are two equivalent ways of defining a list, so both examples should not work(because print has higher priority than [1]):

print ("a", "b")[1];
print qw(a, b)[1];
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But for some reason only the first one does not work...

Your question is yet another example where using strict and warnings makes things a whole lot easier to understand:

print ("a", "b")[1];
print qw(a, b)[1];

__END__

# Result:
syntax error at x.pl line 3, near ")["
Execution of x.pl aborted due to compilation errors.
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But when you add those 2 magic lines at the top:

use strict;
use warnings;

print ("a", "b")[1];
print qw(a, b)[1];

__END__

print (...) interpreted as function at x.pl line 5.
Possible attempt to separate words with commas at x.pl line 6.
syntax error at prog.pl line 5, near ")["
Execution of x.pl aborted due to compilation errors.
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That line about "print (...) interpreted as function ..." is what others (eg. mreece and jdporter) have already discussed.

Another way of looking at it is that:

print ("a", "b")[1];
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is the same, precedence-wise, as:

(print ("a", "b"))[1];
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which doesn't make sense (ie. is a syntax error).  You have to force the desired precedence with:

print (("a", "b")[1]);
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The line "Possible attempt to separate words with commas" is just alerting you to the fact that instead of:

print qw(a, b)[1];
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you probably actually meant something more like:

print qw(a b)[1];
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---

s''(q.S:$/9=(T1';s;(..)(..);$..=substr+crypt($1,$2),2,3;eg;print$..$/

perl -Mdiagnostics -we'print (1,2)[1]'
print (...) interpreted as function at -e line 1 (#1)
    (W syntax) You've run afoul of the rule that says that any list op
+erator
    followed by parentheses turns into a function, with all the list
    operators arguments found inside the parentheses.  See
    perlop/Terms and List Operators (Leftward).

syntax error at -e line 1, near ")["
Execution of -e aborted due to compilation errors (#2)
    (F) Probably means you had a syntax error.  Common reasons include
+:

        A keyword is misspelled.
        A semicolon is missing.
        A comma is missing.
        An opening or closing parenthesis is missing.
        An opening or closing brace is missing.
        A closing quote is missing.

    Often there will be another error message associated with the synt
+ax
    error giving more information.  (Sometimes it helps to turn on -w.
+)
    The error message itself often tells you where it was in the line 
+when
    it decided to give up.  Sometimes the actual error is several toke
+ns
    before this, because Perl is good at understanding random input.
    Occasionally the line number may be misleading, and once in a blue
+ moon
    the only way to figure out what's triggering the error is to call
    perl -c repeatedly, chopping away half the program each time to se
+e
    if the error went away.  Sort of the cybernetic version of S<20
    questions>.

Uncaught exception from user code:
        syntax error at -e line 1, near ")["
Execution of -e aborted due to compilation errors.
 at -e line 1
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Hi,
First off its qw(a b) not qw(a, b)
Yes ("a", "b") and qw(a b) are two ways of defining the same list but saying print ("a", "b")[1] and print qw(a b)[1] are not quite the same!
When you say print ("a", "b")[1] it looks like you're telling Perl "print my list ("a", "b")" and passing it an anonymous array whose single element is 1. That entire construct is confusing to Perl and as such it complains!
However when you say print qw(a b)[1] you're saying to Perl "print the 2nd element of the list qw(a b)", which makes perfect sense to Perl and as such ... it obliges and prints "b" :)
Hope that helps ;)