If the shell scripts start with #!/bin/bash, they will always run with bash from /bin. If they however start with #!/usr/bin/env bash, they will search for bash in $PATH and then start with the first one they can find.
Why would this be useful? Assume you want to run bash scripts, that require bash 4.x or newer, yet your system only has bash 3.x installed and currently your distribution doesn't offer a newer version or you are no administrator and cannot change what is installed on that system.
Of course, you can download bash source code and build your own bash from scratch, placing it to ~/bin for example. And you can also modify your $PATH variable in your .bash_profile file to include ~/bin as the first entry (PATH=$HOME/bin:$PATH as ~ will not expand in $PATH). If you now call bash, the shell will first look for it in $PATH in order, so it starts with ~/bin, where it will find your bash. Same thing happens if scripts search for bash using #!/usr/bin/env bash, so these scripts would now be working on your system using your custom bash build.
One downside is, that this can lead to unexpected behavior, e.g. same script on the same machine may run with different interpreters for different environments or users with different search paths, causing all kind of headaches.
The biggest downside with env is that some systems will only allow one argument, so you cannot do this #!/usr/bin/env <interpreter> <arg>, as the systems will see <interpreter> <arg> as one argument (they will treat it as if the expression was quoted) and thus env will search for an interpreter named <interpreter> <arg>. Note that this is not a problem of the env command itself, which always allowed multiple parameters to be passed through but with the shebang parser of the system that parses this line before even calling env. Meanwhile this has been fixed on most systems but if your script wants to be ultra portable, you cannot rely that this has been fixed on the system you will be running.
It can even have security implications, e.g. if sudo was not configured to clean environment or $PATH was excluded from clean up. Let me demonstrate this:
Usually /bin is a well protected place, only root is able to change anything there. Your home directory is not, though, any program you run is able to make changes to it. That means malicious code could place a fake bash into some hidden directory, modify your .bash_profile to include that directory in your $PATH, so all scripts using #!/usr/bin/env bash will end up running with that fake bash. If sudo keeps $PATH, you are in big trouble.
E.g. consider a tool creates a file ~/.evil/bash with the following content:
#!/bin/bash
if [ $EUID -eq 0 ]; then
echo "All your base are belong to us..."
# We are root - do whatever you want to do
fi
/bin/bash "$@"
Let's make a simple script sample.sh:
#!/usr/bin/env bash
echo "Hello World"
Proof of concept (on a system where sudo keeps $PATH):
$ ./sample.sh
Hello World
$ sudo ./sample.sh
Hello World
$ export PATH="$HOME/.evil:$PATH"
$ ./sample.sh
Hello World
$ sudo ./sample.sh
All your base are belong to us...
Hello World
Usually the classic shells should all be located in /bin and if you don't want to place them there for whatever reason, it's really not an issue to place a symlink in /bin that points to their real locations (or maybe /bin itself is a symlink), so I would always go with #!/bin/sh and #!/bin/bash. There's just too much that would break if these wouldn't work anymore. It's not that POSIX would require these position (POSIX does not standardize path names and thus it doesn't even standardize the shebang feature at all) but they are so common, that even if a system would not offer a /bin/sh, it would probably still understand #!/bin/sh and know what to do with it and may it only be for compatibility with existing code.
But for more modern, non standard, optional interpreters like Perl, PHP, Python, or Ruby, it's not really specified anywhere where they should be located. They may be in /usr/bin but they may as well be in /usr/local/bin or in a completely different hierarchy branch (/opt/..., /Applications/..., etc.). That's why these often use the #!/usr/bin/env xxx shebang syntax.