I have seen many medieval helmets that looked like this:
Why did they have that metal thingy between the eyes? What is the point? To me, it would be harder to see if I had something like that there. It would block my field of vision.
The main reasons are Cost-Efficiency points.
Just consider how expensive is a full covering and well made helmet, and how much they are blocking the view. However this kind of helmet is blocking way less vision, I would say, judging by the position of eyes, not much of the vision will be disturbed. Just close one of your eyes and you can see your nose, switch, you see it from a different angle. If you open both, your nose will not disturb the vision. This works the same, if the eyes have common territory to see, the brain will sum up the sight.
If you want to simulate the effect this part has, put a not too huge mobile phone on position to see how does it look like. Most probably you won't see what is going on in 30-40 centimeters on the front of your face in a line, but you will see what is happening in few meters, and that is the point.
This minor improvement is blocking sword blows from each sides, can stop smaller maces. Instead of causing some serious damage on the face, it has some good chance to get bounced off from the head, and by a good chance if you get this kind of hit, by the force of the blow your head flips backwards, so the weapon is more probably will slip off upwards rather than going straight to your neck.
Note that these helmets were worn by cheaper troops, and since they weren't as important as nobles, it didn't need to provide full defense, but some decent one for low cost.
What it didn't provide is defense against piercing type of weapons. Spears, arrows, etc... only if those weapons hit the metal parts.
If you consider a battle it is not indifferent if you can make a helmet for +10% money to save let's say 30%-40% more blows.
"Medieval" helmets are a very broad category. You're looking at several hundred years of weapon evolution that gets wrapped up into a single term.
In different periods, different people would use a helmet with a nasal. In early medieval times, this would be a significant improvement over a more common simple pot helmet. Not only it protects the nose - the most exposed bit of your face - from direct hits, it also serves (and that might be even more important) as a mounting point for the coif (in later versions).
Coif would be attached to the bottom of the helm in the back and to the nasal in the front, as that provides additional protection for your cheekbones. Also remember that you'd wear a padded cap between your head and the helmet/coif, so that it absorbs the impact of the blow.
Iron being a very premium material in early middle ages, anything that could provide good protection while being significantly cheaper was a huge improvement. The nasal is a very good example of this compromise.
Also, having done some "experimental archaeology" (i.e. actual sword-fighting using 10th-century based weapons), I can tell you that the nasal practically doesn't obstruct your field of vision and makes the crucial difference between a smashed-in nose and just minor scratches.
Why did they have that metal thingy between the eyes? What is the point? To me, it would be harder to see if I had something like that there. it would block my field of vision.
If you have two eyes, it doesn't interfere with vision very much, and you can get used to it.
The point is that many of your opponents would be using swung swords, axes or clubs (or hafts of other weapons that nearly missed with their head) coming at an angle and so, like the hilt of a sword, many such blows might well fall on the nasal and be deflected, instead of blinding you and/or cutting into your face. It could also help against something like a shield bash to the face.
The importance of a nasal is about covering the last remaining uncovered places. It's not usually the only protective gear a fighter would use. If you were wearing such a helmet over an armored hood, most of your entire head, neck and shoulders would be well-protected from most attacks other than a thrust from the front. Compared to the same coif and helmet without a nasal, the nasal might be considered to protect against most attacks that would otherwise encounter no armor, so it's a significant enhancement. Warriors contemporary to nasal helmets (late 8th Century to late 12th Century) would also tend to fight in an armored hauberk and carry a shield that would tend to be near the lower part of the face as well, so aside from a spear jab or arrow, such a fighter was very thoroughly protected. The nasal thus offered important protection for one of the few vulnerable spots remaining.
You're right that nasals like that were added to earlier helmet designs which tended to just be like the helmet shown without the nasal. There weren't many other types of helmet in use when nasals were added, so before about the 13th Century, there wasn't a typical mainstream alternative helmet that was heavier than this, again because the helmet was generally combined with an armored hood.
Probably also to offer some protection to the eyes - a blow from a sword/arrow would hopefully glance or bounce away from the vulnerable area.
The nose guard is kind of a strange item in some ways. They would not provide much protection against a heavy weapon like a mace. I think the main purpose was to prevent getting slashed in the face with a sword.