(This is an expansion of my comment on Kaleb Pederson s answer -- see that answer for a more general treatment of [@] vs [*].)

When bash (or any similar shell) parses a command line, it splits it into a series of "words" (which I will call "shell-words" to avoid confusion later). Generally, shell-words are separated by spaces (or other whitespace), but spaces can be included in a shell-word by escaping or quoting them. The difference between [@] and [*]-expanded arrays in double-quotes is that "${myarray[@]}" leads to each element of the array being treated as a separate shell-word, while "${myarray[*]}" results in a single shell-word with all of the elements of the array separated by spaces (or whatever the first character of IFS is).

Usually, the [@] behavior is what you want. Suppose we have perls=(perl-one perl-two) and use ls "${perls[*]}" -- that s equivalent to ls "perl-one perl-two", which will look for single file named perl-one perl-two, which is probably not what you wanted. ls "${perls[@]}" is equivalent to ls "perl-one" "perl-two", which is much more likely to do something useful.

Providing a list of completion words (which I will call comp-words to avoid confusion with shell-words) to compgen is different; the -W option takes a list of comp-words, but it must be in the form of a single shell-word with the comp-words separated by spaces. Note that command options that take arguments always (at least as far as I know) take a single shell-word -- otherwise there d be no way to tell when the arguments to the option end, and the regular command arguments (/other option flags) begin.

In more detail:

perls=(perl-one perl-two)
compgen -W "${perls[*]} /usr/bin/perl" -- ${cur}

is equivalent to:

compgen -W "perl-one perl-two /usr/bin/perl" -- ${cur}

...which does what you want. On the other hand,

perls=(perl-one perl-two)
compgen -W "${perls[@]} /usr/bin/perl" -- ${cur}

is equivalent to:

compgen -W "perl-one" "perl-two /usr/bin/perl" -- ${cur}

...which is complete nonsense: "perl-one" is the only comp-word attached to the -W flag, and the first real argument -- which compgen will take as the string to be completed -- is "perl-two /usr/bin/perl". I d expect compgen to complain that it s been given extra arguments ("--" and whatever s in $cur), but apparently it just ignores them.

Your title asks about ${array[@]} versus ${array[*]} (both within {}) but then you ask about $array[*] versus $array[@] (both without {}) which is a bit confusing. I ll answer both (within {}):

When you quote an array variable and use @ as a subscript, each element of the array is expanded to its full content regardless of whitespace (actually, one of $IFS) that may be present within that content. When you use the asterisk (*) as the subscript (regardless of whether it s quoted or not) it may expand to new content created by breaking up each array element s content at $IFS.

Here s the example script:

#!/bin/sh

myarray[0]="one"
myarray[1]="two"
myarray[3]="three four"

echo "with quotes around myarray[*]"
for x in "${myarray[*]}"; do
        echo "ARG[*]:  $x "
done

echo "with quotes around myarray[@]"
for x in "${myarray[@]}"; do
        echo "ARG[@]:  $x "
done

echo "without quotes around myarray[*]"
for x in ${myarray[*]}; do
        echo "ARG[*]:  $x "
done

echo "without quotes around myarray[@]"
for x in ${myarray[@]}; do
        echo "ARG[@]:  $x "
done

And here s it s output:

with quotes around myarray[*]
ARG[*]:  one two three four 
with quotes around myarray[@]
ARG[@]:  one 
ARG[@]:  two 
ARG[@]:  three four 
without quotes around myarray[*]
ARG[*]:  one 
ARG[*]:  two 
ARG[*]:  three 
ARG[*]:  four 
without quotes around myarray[@]
ARG[@]:  one 
ARG[@]:  two 
ARG[@]:  three 
ARG[@]:  four 

I personally usually want "${myarray[@]}". Now, to answer the second part of your question, ${array[@]} versus $array[@].

Quoting the bash docs, which you quoted:

The braces are required to avoid conflicts with the shell s filename expansion operators.

$ myarray=
$ myarray[0]="one"
$ myarray[1]="two"
$ echo ${myarray[@]}
one two

But, when you do $myarray[@], the dollar sign is tightly bound to myarray so it is evaluated before the [@]. For example:

$ ls $myarray[@]
ls: cannot access one[@]: No such file or directory

But, as noted in the documentation, the brackets are for filename expansion, so let s try this:

$ touch one@
$ ls $myarray[@]
one@

Now we can see that the filename expansion happened after the $myarray exapansion.

And one more note, $myarray without a subscript expands to the first value of the array:

$ myarray[0]="one four"
$ echo $myarray[5]
one four[5]