The problem is that with the H1n you are stuck with the microphones it has and they are not that great.
With H4N pro ( H4N was the older version) you can connect an external microphone if you have ( up to two external microphones and a stereo input) to get much better sound from your saxophone.
You can also use it to record in 4 channels.*
I have used it as a recorder with good quality microphones and a mixer to record a whole quintet in high quality with excellent results.
The acoustics of the room you record is very important to the final results with any recorder you decide to choose.
Plus the H4n has a wired remote control that is very handy