Yes, it makes a difference. In this case the grid-registered dataset is higher fidelity. However, the cell-registered is easier to handle.
As you see, both types of data are available on the dataset's page. The page states:
The grid-registered is the authoritative registration.
The page also points out:
The cell-registered is derived from the grid-registered,
and the conversion produces slightly flattened relief.
So what's the difference? The grid-registered version puts the data points at the grid corners (the red lines, not the grey squares, define the grid boundaries. You have to imagine the grid cells in the left-hand image). So if there are $n \times m$ grid cells, there are $(n+1) \times (m+1)$ data points in the grid-registered data.
Since there are fewer data points in the cell-registered version, making it involves some smoothing (averaging) of the values in the grid-registered version. There's an illustration of this effect on the other page you linked to; notice how the centre cell of value 90 ends up being smoothed away:
So if accuracy is paramount, you should use the grid-registered version.
You might find cell-registered data more convenient to work with (for me at least it's more intuitively analogous to an image).
The catch is that if you have grid-registered but accidentally treat it as cell-registered, then the data will shift by half a cell. So you should keep careful track of what kind of data you have. As the NOAA page states:
Most grid applications recognize both types and avoid [the grid shift]
problem but care should be taken.
Indeed, one popular open source GIS tool, GRASS, apparently only handles cell-registered data, as explained in the GRASS wiki. So after loading (to wherever) you should check where the pixels of the raster actually end up.