eval "$1" executes the command in the current script. It can set and use shell variables from the current script, set environment variables for the current script, set and use functions from the current script, set the current directory, umask, limits and other attributes for the current script, and so on. bash "$1" executes the command in a completely separate script, which inherits environment variables, file descriptors and other process environment (but does not transmit any change back) but does not inherit internal shell settings (shell variables, functions, options, traps, etc.).

There is another way, (eval "$1"), which executes the command in a subshell: it inherits everything from the calling script but does not transmit any change back.

For example, assuming that the variable dir isn't exported and $1 is cd "$foo"; ls, then:

• cd /starting/directory; foo=/somewhere/else; eval "$1"; pwd lists the content of /somewhere/else and prints /somewhere/else.
• cd /starting/directory; foo=/somewhere/else; (eval "$1"); pwd lists the content of /somewhere/else and prints /starting/directory.
• cd /starting/directory; foo=/somewhere/else; bash -c "$1"; pwd lists the content of /starting/directory (because cd "" doesn't change the current directory) and prints /starting/directory.

eval "$1" executes the command in the current script. It can set and use shell variables from the current script, set environment variables for the current script, set and use functions from the current script, set the current directory, umask, limits and other attributes for the current script, and so on. bash -c "$1" executes the command in a completely separate script, which inherits environment variables, file descriptors and other process environment (but does not transmit any change back) but does not inherit internal shell settings (shell variables, functions, options, traps, etc.).

There is another way, (eval "$1"), which executes the command in a subshell: it inherits everything from the calling script but does not transmit any change back.

For example, assuming that the variable dir isn't exported and $1 is cd "$foo"; ls, then:

• cd /starting/directory; foo=/somewhere/else; eval "$1"; pwd lists the content of /somewhere/else and prints /somewhere/else.
• cd /starting/directory; foo=/somewhere/else; (eval "$1"); pwd lists the content of /somewhere/else and prints /starting/directory.
• cd /starting/directory; foo=/somewhere/else; bash -c "$1"; pwd lists the content of /starting/directory (because cd "" doesn't change the current directory) and prints /starting/directory.