Would there be a difference in military skill/manpower or gender ratio between a country that has a mandatory draft for men and a country where military is completely voluntary? I’m completely aware there are a lot of other reasons for differences but what differences might tie in due to this particular reason?
This one is fairly simple with a few implications.
So, compulsory military service will have a larger number of troops available, than services that rely on voluntary enlistment. The numbers can get a little complicated when you consider that compulsory enlistment will (usually) result in shorter enlistment times (during peacetime), while voluntary enlistment is likely to result in personnel who remain active for longer (again, during peacetime.)
During wartime, drafts can push your overall numbers up, significantly. So, in nations with compulsory service, it’s not uncommon to see very large chunks of the population technically existing as reserve forces who can be called up if needed.
As for which is better? That’s much harder to quantify.
In theory, a force relying on voluntary enlistment will produce better soldiers, but in smaller numbers. In practice, things get a lot more complicated, and this theory hasn’t always been true historically.
Volunteer militaries are (usually) cheaper to operate. You need to spend less to equip and train them. (Turnover will be lower, which reduces training costs overall. You don’t have to worry about policing draft dodging, and desertion rates will be lower.)
One time this really flips is during prolonged, unpopular wars. In those, voluntary forces will usually choose to muster out their term of service is up rather than reenlisting. This means you need to spend more on training, and also need to increase pay to keep the military attractive as a job. This can create situations where a conscripted force would actually be cheaper. Though, is a problem where you’d have soldiers who were only there because they were forced to, and would desert at the first opportunity.)
Finally, there are hybrid systems, where you have a corps of soldiers who joined of their own volition, and a large pool of drafted soldiers filling out the ranks. If it’s technically unified, then it will look more like a conscripted force, with a few individuals who want to be there mixed in. Though, there are other potential situations, including special forces (where the main force is conscripted, but some of the specialist groups are there by choice.)
When you isolate cultural issues, voluntary military service will closely align with overall social demographics. You can see this in the US military when looking at racial demographics. At the same time, the US military is predominantly male, however, there are significant cultural considerations on that topic. There are examples of other nations, with compulsory service, that maintain gender parity in their armed forces, so it is theoretically possible (independent of cultural issues) for a volunteer force to also reach full parity.
If you have a conscripted standing force, it will probably be larger, but lower quality. If you have a volunteer force, it will be smaller, but the individual soldiers will be more effective. If either military restricts their demographics, that will harm their ability to find recruits. Technically, a conscripted force will probably work towards filling to a fixed quota, so, restricting conscription of women may have no effect on the overall size, but reduce the quality, by opening the door for recruits that would have washed out if better candidates were available. This is also true with volunteer force, where the more recruits you have to choose from, the pickier you can be.
Somewhat obviously, if one nation prohibits women from serving, and the other does not, then the latter will have more women in their armed forces.
So, in either case, the more people you have to pick from, the better your options will be.