Why does `watch` make `ls /tmp` list contents of $HOME?

The difference may be seen via strace:

$ strace -ff -o bq watch sh -c 'ls\ /tmp/|wc -l'
^C
$ strace -ff -o nobq watch sh -c 'ls /tmp/|wc -l'
^C
$ grep exec bq* | grep sh
bq.29218:execve("/usr/bin/watch", ["watch", "sh", "-c", "ls\\ /tmp/|wc -l"], [/* 54 vars */]) = 0
bq.29219:execve("/bin/sh", ["sh", "-c", "sh -c ls\\ /tmp/|wc -l"], [/* 56 vars */]) = 0
bq.29220:execve("/bin/sh", ["sh", "-c", "ls /tmp/"], [/* 56 vars */]) = 0
$ grep exec nobq* | grep sh
nobq.29227:execve("/usr/bin/watch", ["watch", "sh", "-c", "ls /tmp/|wc -l"], [/* 54 vars */]) = 0
nobq.29228:execve("/bin/sh", ["sh", "-c", "sh -c ls /tmp/|wc -l"], [/* 56 vars */]) = 0
nobq.29229:execve("/bin/sh", ["sh", "-c", "ls", "/tmp/"], [/* 56 vars */]) = 0

In the backquote case, ls /tmp is passed as a single argument to the -c to sh, which runs as expected. Without this backquote, the command is instead word split when watch runs sh which in turn runs the supplied sh, so that only ls is passed as the argument to -c, meaning that the sub-subsh will only run a bare ls command, and lists the contents of the current working directory.

So, why the complication of sh -c ...? Why not simply run watch 'ls /tmp|wc -l' ?

There are two main categories of watch commands (of the ones that are to run commands periodically, watch is not a standard command, there are even systems where watch does something completely different like snooping on another tty line on FreeBSD).

One that already passes the concatenation of its arguments with spaces to a shell (it does in effect call sh -c <concatenation-of-arguments>) and one that just runs the command specified with the arguments specified without invoking a shell.

You're in the first situation, so you just need:

watch 'ls /tmp/|wc -l'

When you do:

watch sh -c 'ls /tmp/|wc -l'

your watch actually runs:

sh -c 'sh -c ls /tmp/|wc -l'

And sh -c ls /tmp/ is running the ls inline script where $0 is /tmp/ (so ls is run without arguments and lists the current directory).

Some of the watch implementations in the first category (like the one from procps-ng on Linux) accept a -x option to make them behave like the watch of the second category. So with there, you can do:

watch -x sh -c 'ls /tmp/|wc -l'