What is the difference between "chmod +x" and "chmod 755"?

Short version:

To be able to compare them, we should look at them from the same perspective, so:

• chmod +x is equal to chmod ugo+x (Based on umask value)
• chmod 755 is equal to chmod u=rwx,go=rx

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Explanation:

Firstly you should know that:

1. + means add this permission to the other permissions that the file already has.

2. = means ignore all permissions, set them exactly as I provide.

   • So all of the "read, write, execute, sticky bit, suid and guid" will be ignored and only the ones provided will be set.

3. read = 4, write = 2, execute = 1

   • Here is the binary logic behind it (if you're interested):

     Symbolic:  r-- -w- --x  |  421
     Binary:    100 010 001  |  -------
     Decimal:    4   2   1   |  000 = 0
                             |  001 = 1
     Symbolic:  rwx r-x r-x  |  010 = 2
     Binary:    111 101 101  |  011 = 3
     Decimal:    7   5   5   |  100 = 4
                /   /   /    |  101 = 5
     Owner  ---/   /   /     |  110 = 6
     Group  ------/   /      |  111 = 7
     Others ---------/       |  Binary to Octal chart
     

Using +x you are telling to add (+) the executable bit (x) to the owner, group and others.

• it's equal to ugo+x or u+x,g+x,o+x
• When you don't specify which one of the owner, group or others is your target, in case of x it will considers all of them. And as @Rinzwind pointed out, it's based on umask value, it adds the bit to the ones umask allows. remember if you specify the target like o+r then umask doesn't have any effect anymore.
• It doesn't touch the other mods (permissions).
• You could also use u+x to only add executable bit to the owner.

Using 755 you are specifying:

• 7 --> u=rwx (4+2+1 for owner)
• 5 --> g=rx (4+1 for group)
• 5 --> o=rx (4+1 for others)

So chmod 755 is like: chmod u=rwx,g=rx,o=rx or chmod u=rwx,go=rx.

chmod +x adds the execute permission for all users to the existing permissions.

chmod 755 sets the 755 permission for a file.

755 means full permissions for the owner and read and execute permission for others.

Another way to look at it (which I find easier to understand) is that chmod +x is setting the permissions relatively, whereas chmod 755 is setting them absolutely.

After chmod 755 is ran on a file, its permissions will be 755, or rwxr-xr-x.

chmod +x will just take the existing permissions, and add execute permissions to the file.