Select controller:Expert mode
Get medals on every level to unlock Expert mode. In this mode the units are tougher and harder which means enemies that were safe enemies are bound to attack on sight. You also lose wings easier and you also lose health faster. On the plus side, Fox can now wear sunglasses like his father in Venom 2.
New title screen
Get medals on every level in Expert mode to unlock a new title screen.
Different ending
Although subtle, if you let a missile hit Great Fox in Sector Z and finish the game, then when Star Fox is running as the credits roll and Great Fox appears from the horizon, the wing that was destroyed will still appear.
64 logo
On the menu where it tells you press 
and shows Star Fox, if you move 
(you don't have to hold a button on the N64 cartridge) it will move the 64 symbol around.
Play as characters in versus
If you get the medal on Venom on hard mode you unlock being able to play versus without vehicles and instead play as the actual characters wielding bazookas.