It's called the D-pad. I prefer the D-pad for side scrollers and menu based games, and the analog stick for 3D and birdseye view games. Assuming it allows 360 degrees of movement, unlike the original Zelda.
yeah it really depends on what the character is doing. I was playing pokemon diamond on my 3ds yesterday, and I noticed, the joystick felt really weird controlling a character that can only move in 4 directions. The advantage of a joystick is that it has a crapton of directions, so there's a lot more freedom of movement.
As a fellow iDevice and Android game developer I must recommend you use a circle pad, while they are a bit trickier to code (at least it was for me) they are much more responsive and easier to use for the average iOS and Android gamer. Virtual D-Pads are usually always hard to use and unresponsive.
And despite the fact that your game only uses four directions, if you plan on working on future games that would allow your character to move in more than it's an easy matter of just recycling the code then having to start all over by making a circle pad then.
I feel like the best games for touch screen based phones are the ones that are specifically developed around their strengths (accelerometers, capacitive touch, etc), and don't try to shoehorn ordinary game control mechanics in.
I would almost recommend developing your game around these features and limitations, rather than using virtual sticks/d-pads which are, in my opinion, unintuitive.