What is the difference between the functions of the exec family of system calls like exec and execve?

There is no exec system call -- this is usually used to refer to all the execXX calls as a group. They all do essentially the same thing: loading a new program into the current process, and provide it with arguments and environment variables. The differences are in how the program is found, how the arguments are specified, and where the environment comes from.

• The calls with v in the name take an array parameter to specify the argv[] array of the new program. The end of the arguments is indicated by an array element containing NULL.

• The calls with l in the name take the arguments of the new program as a variable-length argument list to the function itself. The end of the arguments is indicated by a (char *)NULL argument. You should always include the type cast, because NULL is allowed to be an integer constant, and default argument conversions when calling a variadic function won't convert that to a pointer.

• The calls with e in the name take an extra argument (or arguments in the l case) to provide the environment of the new program; otherwise, the program inherits the current process's environment. This is provided in the same way as the argv array: an array for execve(), separate arguments for execle().

• The calls with p in the name search the PATH environment variable to find the program if it doesn't have a directory in it (i.e. it doesn't contain a / character). Otherwise, the program name is always treated as a path to the executable.

• FreeBSD 5.2 added another variant: execvP (with uppercase P). This is like execvp(), but instead of getting the search path from the PATH environment variable, it's an explicit parameter to the function:

int execvP(const char *file, const char *search_path, char *const argv[]);

Use man exec and read:

The execv(), execvp(), and execvpe() functions provide an array of pointers to 
null-terminated strings that represent the argument list available to the new program. 
The first argument, by convention, should point to the filename associated with the file 
being executed. The array of pointers must be terminated by a NULL pointer. 

execv

int execv(const char *path, char *const argv[]);

So you pass an array as parameters

int execle(const char *path, const char *arg,
              ..., char * const envp[]);

Almost the same, but not as an array, but rather as a list of values (strings), followed by an array the designates the environment.

Here:

int execvp(const char *file, char *const argv[]);

You are calling a file, without path, so it expects you to be already in the right path before calling.

Last but not least:

int execve(const char *filename, char *const argv[],
                  char *const envp[]);

Similar to previous one, but now you have two arrays, for arguments and environment variables.

Since all of these function belongs to exec() family, let me differentiate according to extra characters with the meanings,

1.execve():

p : not present => name of the program to run will be taken from pathname

v : present => argument will be passed as array

e : present => environment will be taken from envp argument

2.execle():

p : not present => name of the program to run will be taken from pathname

l : present => argument will be passed as list

e : present => environment will be taken from envp argument

3.execlp():

p : present => name of the program to run will be taken from filename specified or system will search for program file in PATH variable.

l : present => argument will be passed as list

e : not present => environment will be taken from caller's environ

4.execvp():

p : present => name of the program to run will be taken from filename specified or system will search for program file in PATH variable.

v : present => argument will be passed as array

e : not present => environment will be taken from caller's environ

5.execv():

p : not present => name of the program to run will be taken from pathname

v : present => argument will be passed as array

e : not present => environment will be taken from caller's environ

6.execl():

p : not present => name of the program to run will be taken from pathname

l : present => argument will be passed as list

e : not present => environment will be taken from caller's environ