Difference between string.h and cstring?

In C++ you should include cstring as the header while in c you should include string.h as the header.

In C++

#include <cstring>

In C

#include <string.h>

Features of C standard Library are also provided in the C++ Standard library and as a general naming convention they are pre-pended by an c to the corresponding names in C standard library.

For Example:
string.h becomes cstring
stdio.h becomes cstdio and so on…

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Since other answers have added different dimensions to this discussion,I felt compelled to refer the holy standard to clear this bit.

As per C++11 20.9.14.6 & 7:

Table 55 describes the header <cstring>.
The contents are the same as the Standard C library header , with the change to memchr() specified in 21.7.

While 21.7 Null-terminated sequence utilities states:

The function signature memchr(const void*, int, size_t) shall be replaced by the two declarations:

const void* memchr(const void* s, int c, size_t n);
void* memchr( void* s, int c, size_t n);

both of which shall have the same behavior as the original declaration.

Annex D (normative) Compatibility features [depr] states:

D.6 C standard library headers

1 For compatibility with the C standard library and the C Unicode TR, the C++ standard library provides the 25 C headers, as shown in Table 151.

Which include:

<assert.h> <float.h> <math.h> <stddef.h> <tgmath.h>
<complex.h> <inttypes.h> <setjmp.h> <stdio.h> <time.h>
<ctype.h> <iso646.h> <signal.h> <stdint.h> <uchar.h>
<errno.h> <limits.h> <stdarg.h> <stdlib.h> <wchar.h>
<fenv.h> <locale.h> <stdbool.h> <string.h> <wctype.h>

Further on,

2 Every C header, each of which has a name of the form name.h, behaves as if each name placed in the standard library namespace by the corresponding cname header is placed within the global namespace scope. It is unspecified whether these names are first declared or defined within namespace scope (3.3.6) of the namespace std and are then injected into the global namespace scope by explicit using-declarations (7.3.3).

3 [ Example: The header <cstdlib> assuredly provides its declarations and definitions within the namespace std. It may also provide these names within the global namespace. The header <stdlib.h> assuredly provides the same declarations and definitions within the global namespace, much as in the C Standard. It may also provide these names within the namespace std. —end example ]

Conclusion:

From the above references:
I stand corrected on my earlier suggestion, there seems to be no apparent advantage of using cstring over string.h while as @Alf suggested there might be some compilation issues due to use of unqualified function names when using cstring as header. So given hat there is no apparent dis-advantage of using string.h or advantage of using cstring, I think either can be used in C++ if used in a proper manner.

There is a subtle difference between string.h and cstring

Answer of Alf P. Steinbach (can be found as a comment to the asked question):

string.h places the identifiers in the global namespace, and may also place them in the standard namespace. While cstring places the identifiers in the standard namespace, and may also place them in the global namespace. You definitely don’t want that cstring behavior, because code that e.g. uses just strlen may work fine with one compiler, then fail to compile with another compiler. It’s very unpleasant surprise. So for C and C++, use the more safe string.h.