There's nothing specific to asm here; it's the same issue in C. It's all about how you store strings in memory and keep track of where they end.
what is the different between a null terminated string and a string that is not terminated by null?
A null-terminated string has a 0 byte after it, so you can find the end with strlen. (e.g. with a slow repne scasb). This makes is usable as an implicit-length string, like C uses.
NASM Assembly - what is the ", 0" after this variable for? explains the NASM syntax for creating one in static storage with db. db usage in nasm, try to store and print string shows what happens when you forget the 0 terminator.
Are they interchangeable?
If you know the length of a null-terminated string, you can pass pointer+length to a function that wants an explicit-length string. That function will never look at the 0 byte, because you will pass a length that doesn't include the 0 byte. It's not part of the string data proper.
But if you have a string without a terminator, you can't pass it to a function or system-call that wants a null-terminated string. (If the memory is writeable, you could store a 0 after the string to make it into a null-terminated string.)
In Linux, many system calls take strings as C-style implicit-length null-terminated strings. (i.e. just a char* without passing a length).
For example, open(2) takes a string for the path: int open(const char *pathname, int flags); You must pass a null-terminated string to the system call. It's impossible to create a file with a name that includes a '\0' in Linux (same as most other Unix systems), because all the system calls for dealing with files use null-terminated strings.
OTOH, write(2) takes a memory buffer which isn't necessarily a string. It has the signature ssize_t write(int fd, const void *buf, size_t count);. It doesn't care if there's a 0 at buf+count because it only looks at the bytes from buf to buf+count-1.
You can pass a string to write(). It doesn't care. It's basically just a memcpy into the kernel's pagecache (or into a pipe buffer or whatever for non-regular files). But like I said, you can't pass an arbitrary non-terminated buffer as the path arg to open().
Or they are not equivalent of each other?
Implicit-length and explicit-length are the two major ways of keeping track of string data/constants in memory and passing them around. They solve the same problem, but in opposite ways.
Long implicit-length strings are a bad choice if you sometimes need to find their length before walking through them. Looping through a string is a lot slower than just reading an integer. Finding the length of an implicit-length string is O(n), but an explicit-length string is of course O(1) time to find the length. (It's already known!). At least the length in bytes is known, but the length in Unicode characters might not be known, if it's in a variable-length encoding like UTF-8 or UTF-16.