PHP, 74 bytes

<?=long2ip($i=ip2long($argv[1])&$m=ip2long($argv[2])),' ',long2ip($i|~$m);

As a standalone, input is via command line:

$ php ip.php 192.168.0.1 255.255.255.0
192.168.0.0 192.168.0.255

Try it online!

Or as a function, 80 bytes:

function($a,$b){return[long2ip($i=ip2long($a)&$m=ip2long($b)),long2ip($i|~$m)];}

Try it online!

Ungolfed

function ip( $a, $b ) {
    $i = ip2long( $a );          // string IP to 32 bit int
    $m = ip2long( $b );          // string netmask to 32 bit int
    $n = $i & $m;                // network is bitwise AND of IP and netmask
    $c = $i | ~$m;               // broadcast is bitwise OR of IP and inverse netmask
    return [ long2ip( $n ), long2ip( $c ) ];
}

PHP has nice (albeit with long function names) built-ins to handle IPv4 dotted-string to binary and back.

Output

192.168.0.1 255.255.255.0   => 192.168.0.0 192.168.0.255
192.168.0.0 255.255.255.0   => 192.168.0.0 192.168.0.255
192.168.0.255 255.255.255.0 => 192.168.0.0 192.168.0.255
100.200.100.200 255.255.255.255 => 100.200.100.200 100.200.100.200
1.2.3.4 0.0.0.0 => 0.0.0.0 255.255.255.255
10.25.30.40 255.252.0.0 => 10.24.0.0 10.27.255.255

Factor, 103 bytes

[ [ ipv4-aton ] bi@ 2dup bitand -rot dup bit-count 32 - abs on-bits pick bitor 2nip [ ipv4-ntoa ] bi@ ]

Nice.

Ungolfed:

: mask-and-broadcast ( ip mask -- netaddr broadcast )
  [ ipv4-aton ] bi@ 2dup bitand -rot dup bit-count 32 - abs on-bits pick bitor 2nip
  [ ipv4-ntoa ] bi@ ;

JavaScript (ES6), 92 bytes

(a,m)=>a.split`.`.map((n,i)=>(v=m[i],m[i]=n&v|v^255,n&v),m=m.split`.`).join`.`+" "+m.join`.`

Explanation

(a,m)=>
  a.split`.`
  .map((n,i)=>(
      v=m[i],
      m[i]=n&v|v^255,
      n&v
    ),
    m=m.split`.`
  ).join`.`
  +" "+m.join`.`

Test

var solution = (a,m)=>a.split`.`.map((n,i)=>(v=m[i],m[i]=n&v|v^255,n&v),m=m.split`.`).join`.`+" "+m.join`.`

<input type="text" id="input" value="10.25.30.40 255.252.0.0" />
<button onclick="result.textContent=solution.apply(null,input.value.split` `)">Go</button>
<pre id="result"></pre>

Perl, 90 85 bytes

includes +6 for -pF/\D/

for(0..3){push@a,$F[$_]&1*($y=$F[$_+4]);push@b,$F[$_]|~$y&255}$"='.';$_="@a @b"

Usage:

echo "192.168.0.1 255.255.255.0" | perl -pF/\\D/ file.pl

More readable:

for(0..3) {
    push @a, $F[$_] & 1*($y=$F[$_+4]);  # calc/add network address 'byte'
    push @b, $F[$_] | ~$y & 255         # calc/add broadcast address 'byte'
}
$"='.';                                 # set $LIST_SEPARATOR
$_="@a @b"                              # set output to network address and broadcast

The -F/\D/ splits the input on non-digits and stores it in @F.

PHP, 126 bytes

With input in $n:

preg_filter(~Ð×£ÔÖÐ,~Û¤¢Â×ÛÎÖÑ×ÛÔÔÁÌÀ×ÛÑÂ×ÛÂÛÁÊÀÝÑÝÅÝÝÖÑ×Û¤ÛÒÊ¢ÙÛÎÖÖÑ×ÛÑÂÛÑ×Û¤ÛÒÊ¢ÍÊÊÙÛÎÖÖÅÝÝÖ,$n);echo"$c $b";

Hexdump:

0000000: 7072 6567 5f66 696c 7465 7228 7ed0 d7a3  preg_filter(~...
0000010: 9bd4 d6d0 9a2c 7edb 9ea4 a2c2 d7db ced6  .....,~.........
0000020: d1d7 db96 d4d4 c1cc c0d7 db9c d1c2 d7db  ................
0000030: 8bc2 db96 c1ca c0dd d1dd c5dd ddd6 d1d7  ................
0000040: db9e a4db 96d2 caa2 d9db ced6 d6d1 d7db  ................
0000050: 9dd1 c2db 8bd1 d7db 9ea4 db96 d2ca a283  ................
0000060: cdca cad9 81db ced6 d6c5 dddd d62c 246e  .............,$n
0000070: 293b 6563 686f 2224 6320 2462 223b       );echo"$c $b";

And a more readable version:

preg_filter( /* PCRE regex on input */
    '/(\d+)/e', /* match all digits, execute the code for each one */
    '$a[]=($1) /* push the matched value to the array $a */
        .($i++>3 /* if we're at the 5th or higher digit */
            ?($c.=($t=$i>5?".":"").($a[$i-5]&$1)) /* append to $c bitwise AND-ed number */
                .($b.=$t.($a[$i-5]|255&~$1)) /* append to $b the broadcast address */
            :"")',
    $n);
echo"$c $b"; /* output $c and $b */

preg_filter requires a single statement in the replacement pattern when using the e flag, so I 'append' the result of the calculations to the 5th and higher values of $a, because those are never reused.

Pyth, 44 42 bytes

LisMcb\.256Lj\..[04jb256'J.&yzKyw'+JxKt^2 32

Try it online. Test suite.

Expects the input as an array, like ["10.25.30.40", "255.252.0.0"].

MATL, 47 bytes

This answer uses current version (4.0.0) of the language.

'%i.%i.%i.%i't32whh2:"j'\d+'XXU]tbZ&tb255Z~+hYD

Example

>> matl
 > '%i.%i.%i.%i't32whh2:"j'\d+'XXU]tbZ&tb255Z~+hYD
 > 
> 192.168.0.1
> 255.255.255.0
192.168.0.0 192.168.0.255

Explanation

'%i.%i.%i.%i't32whh      % format string: '%i.%i.%i.%i %i.%i.%i.%i'
2:"                      % for loop: do this twice
    j'\d+'XXU            % input string and parse into 4-vector with the numbers
]                        % end
tbZ&                     % compute network address
tb255Z~+                 % compute broadcast address
hYD                      % concatenate into 8-vector and apply format string