$ make install
This will install the tool into $HOME/bin. After that you can use the 'diff' sub-option to find functions that changed between the two versions of the assembly file. An invocation could look like this:
$ asmtool diff file_old.s file_new.s
This gives you a list of added, removed, and changed functions. To see what changed the functions, use the -s option:
$ asmtool diff -s file_old.s file_new.s
There are more options available to control the diff output. Use
$ asmtool diff --help
to get an overview.