Ruby, 11

s = 'string'
t = 'string'
!!s[t]&t[s]

Checks if each string is contained within the other.

JavaScript, 11

Strings have to be stored in a and b.

!(a>b||a<b)

Python 49 45 18 22 15

( + 3 if string variables are considered)

print {a:0,b:1}[a]

The string should be hard coded at the two occurrences of a and one occurrence of b surrounded by quotes.

a and b should be per-initialized to the strings.

Python shell, 9

( + 3 if string variables are considered)

{a:0,b:1}[a]

Output in shell

>>> a = 'string'
>>> b = 'string'
>>> {a:0,b:1}[a]
1
>>> a = 'string'
>>> b = 'stringgg'
>>> {a:0,b:1}[a]
0
>>> {'string':0,'string':1}['string']
1
>>> {'stringggg':0,'string':1}['stringggg']
0
>>> 

Explanation

Creates a dict(hash table) with the key of first and second string. If second string is the same, the value of first is replaced by that of second. Finally, we print the value of first key.

EDIT: OP allowed 0/1 instead of False/True as well as using per-initialized variables.

GolfScript (5 chars)

'string1''string1'].&,(

Fairly straightforward port of the PHP reference implementation. Leaves 0 (=false) on the stack if the strings are the same, or 1 (=true) if they're different.

coreutils: uniq -d

Just enter your two strings as the standard input of a pipe and uniq -d | grep -q . will print nothing but will have a return value of success or error. If you want to print the boolean, just replace with uniq -d | grep -c .

How many characters? I let you count; uniq -d|grep -q . with no extra spaces has 17 characters in it, but since the whole job is performed by uniq, I would say this solution is a 0-character one in... uniq own language!

Actually, uniq -d will print one line if the two strings are identical, and nothing if the are different.

C++, 63 58

const char* a = "string";
const char* b = "string";
int main(){while(*a&&*b&&!(*a^*b))++a,++b;return!(*a^*b);}

Python, 79

a=list(a)
b=list(b)
for i in range(len(a)):
 if ord(a[i])-ord(b[i])<0:
  print 'n'

Python - 52 chars without strings (76 with strings)

The in operator might be skirting the edges, but the rules disallow strict equality comparison, not similarity comparison.

a="string"
b="string"
x=a in b and not(len(a)-len(b)>0 or len(a)-len(b)<0)

grep 14 characters

Of course, I only count the grep code; the two strings are on two consecutive lines in the input (either a pipe or a file or even an interactive session).

$ echo -e 'string\nstring' | grep -cPzo "(?s)^(\N*).\1$"
1
$ echo -e 'string\nstring1' | grep -cPzo "(?s)^(\N*).\1$"
0
$ echo -e 'string1\nstring' | grep -cPzo "(?s)^(\N*).\1$"
0

Matlab: 12 chars (after the strings are in variables)

~(x*x'-y*y')

The code including assignments would be:

x='string1'
y='string2'
~(x*x'-y*y')

Perl, 24 bytes

$r=!(-1+keys{$a,1,$b,1})

The two strings are given in $a and $b. The result is stored in $r.

The trick is using a hash array. If the strings are equal, then only one hash entry is created.

A standalone variant that gets the string from STDIN and returns 0, if the strings match and 1 otherwise:

$h{<>}=$h{<>}=1;exit -1+keys%h

The very crazy way

Just for the fun, but many ways for making it fail if one thinks about it. More over, don't forget the strings will be EXECUTED by the shell.

$ echo -e 'string\nstring1' | sed -e '1s/^/#define /' | cpp | sh 2>/dev/null && echo true
$ echo -e 'string\nstring' | sed -e '1s/^/#define /' | cpp | sh 2>/dev/null && echo true
true
$ echo -e 'string1\nstring' | sed -e '1s/^/#define /' | cpp | sh 2>/dev/null && echo true

A good counter-example is comparing "string" as first string and "rm -Rf /" as a second string; just check as root and see: it will say "true" though both strings obviously aren't the same.

Python 2 - 50 (including strings)

a='string'
b='string'
len(''.join(set(a+b)))==len(a)

Output

True

Non-equal strings

a='string'
b='String'
len(''.join(set(a+b)))==len(a)

Output

False

Ruby, 21

Online version

Strings have to be stored in a and b:

!!(a+b=~/^(#{a})\1$/)

Checks if the concatenation of a and b contains a twice.

D 30

string a="somestring";
string b="somestring";

return !reduce!"a|b"(a[]^b[]);

APL (8)

{∧/⍺∊¨⍵}

• ⍺∊¨⍵: for each combination of elements in ⍺ and ⍵, see if the element from ⍺ contains the element from ⍵. Since in a string these will all be single characters, and a string contains itself, this is basically comparing each pair of characters.
• ∧/: take the logical and of all the values (if all the characters match, the strings are equal)

C/C++ 43

bool foo()
{
    char *a="string1", *b="string2";
    for(;!(*a-*b)&&*a++&&*b++;);return!(*a+*b);
}

Python 15 - 2( if you hard code the string in the solution) = 13

>>> s = 'string'
>>> t = 'string'
>>> 1 is len({s,t})
True
>>> s = 'string1'
>>> t = 'string2'
>>> 1 is len({s,t})
False

Python 17

>>> s = 'string'
>>> t = 'string'
>>> s in t and t in s
True
>>> s = 'string1'
>>> t = 'string2'
>>> s in t and t in s
False

Python - 11 (without the strings)

>>> a = 'ss'
>>> b = 's'
>>> a in b in a
False

Perl, 25 bytes

$r="$a$b"=~/^(\Q$a\E)\1$/

The input is expected in variables $a and $b. The result is stored in $r. It uses a regular expression to check, whether the concatenation "$a$b" is a duplicate of $a.

C#

var b="string";
var a="string";
return (a.Length==b.Length)&&Regex.IsMatch(a, b);