During my career, I probably made several thousand glass melts. Since I was interested in the science of glass, I always started with a series of glasses using mol% compositions. In order to calculate a batch, I had to convert to wt% compositions. Then, since many of the batch components were not in the same chemical form as that in my desired glass, I had to also take in to account the loss of gases on melting. For example, I might set up my series of compositions based on sodium oxide in the glass. I would then convert the composition to wt% and further take into account that my sodium oxide would actually be added to the batch as sodium carbonate, sodium nitrate, or some other compound. (You can't actually add alkali oxides directly since they aren't usually available in that form.) In other cases, e.g. silica, there is no correction to that calculated from the wt% needed since the batch material is also silica. It gets even more complicated when using batch materials which supply two or more desired oxides, e.g. using a feldspar which contains sodium, aluminum, and silicon to form a sodium aluminosilicate glass with a different composition from that of the feldspar. See my text Introduction to Glass Science and Technology for a thorough explanation of batching.
This is common for many practical application since Alan already pointed out. For other applications you need to know how many atoms are considered, e.g. in order to get information about the composition. Fortunately you can convert these information into each other.
There is some more background. The practical aspect has already been mentioned. But the value of "weight%" becomes obvious if you do chemical analysis (to quantify the compsition of your glass or anything else). In order to know the weight fraction (that is the actual quantity, it may be dimensionless or have the unit %) of a component of your material, you only have to perform the analysis for that component. If you want to know the molar fraction (again dimensionless qunatity, but you can use mol%) of the same component you have to know all the other components of your mixture. That is a big difference if you do not know EVERYTHING. Moreover, if you have molar fraction values or molar quantities (molar formation energy or so) you have to be very sure, to which formula unit (Al2O3 or AlO1.5 or Al1/3O2/3) that quantity refers to (and often it is not indicated clearly!). It is no problem if your quantities refer to mass of the component! Although coming from natural sciences the molar/atomic quantities look more obvious, mass-related quantities can help you to avoid a lot of trouble.
Atomic percent is based on the number of atoms in a sample. So if the sample has x number of oxygen atoms and x number of iron atoms it would report 50% oxygen and 50% iron (atomic percent). Weight percent is based on the mass of the elements detected. So if we used the above example and reported the results as weight percent we would get 22.3% oxygen and 77.7% iron. Weight percent takes into consideration the mass or atomic weight of the elements and not just the number of atoms. Most people use weight percent although for chemistry atomic percent may be more useful. (transferred)
During my career, I probably made several thousand glass melts. Since I was interested in the science of glass, I always started with a series of glasses using mol% compositions. In order to calculate a batch, I had to convert to wt% compositions. Then, since many of the batch components were not in the same chemical form as that in my desired glass, I had to also take in to account the loss of gases on melting. For example, I might set up my series of compositions based on sodium oxide in the glass. I would then convert the composition to wt% and further take into account that my sodium oxide would actually be added to the batch as sodium carbonate, sodium nitrate, or some other compound. (You can't actually add alkali oxides directly since they aren't usually available in that form.) In other cases, e.g. silica, there is no correction to that calculated from the wt% needed since the batch material is also silica. It gets even more complicated when using batch materials which supply two or more desired oxides, e.g. using a feldspar which contains sodium, aluminum, and silicon to form a sodium aluminosilicate glass with a different composition from that of the feldspar. See my text Introduction to Glass Science and Technology for a thorough explanation of batching.
I have to say that I have difficulties with expressing any compositions with materials that do not exist in that form in the material e.g. Na2O, CaO. You can't weigh these ingredients out easily. Obviously, as hinted before, these are reactive species with water. I think using this form can only serve to confuse and is, perhaps, a relic of the past. However, I see the issues in quoting Na (as opposed to Na+) when again this is a reactive species. Glass is not made up of a collection or mixture of reactive oxides - but reacted oxides - otherwise it would be entertaining to plunge a glass into water when washing up...
concerning Na2O etc: At least in thermodynamics of oxides these "oxides" are used as components in the sense of Gibbs phase rule. You reconstruct phase diagrams from these components instead from atomic components (as you do it in alloy thermodynamics). In this way a phase diagram Al-Mg-O can be simplified in terms of a pseudo-binary section MgO-Al2O3. Similar argument can be used in chemical analysis in oxides, or even in materials containing partially complex ions like (SO4)2-. The you describe it in terms of Na2O + SO3. Slag people do it in this way. The point may really be you have to deal with one component (O) less.
I understand your point, Andreas, but describing sodium sulfate as Na2O plus SO3 seems to me (and I'm sure to others) as particularly confusing unless one is looking at the enthalpy of formation and Gibbs Free energy changes (as you say with thermodynamics). Why not try Na2S + 2 O2? Sodium sulfate bears no relation (other than containing the elements) to Na2O or SO3 in the same way that water bears no resemblance to hydrogen and oxygen. I think there has to be a better way forward than these routes.
However, in a practical sense, the Gladstone-Dale 'rule' works well when one considers certain minerals as mixes of the oxides and the prediction of RI is surprisingly good in many cases.