What are the advantages/disadvantages of going either route? From what I can tell, they have about the same number of missions, and the cost of the three Gemini missions combined costs about as much as the Apollo project. That said, not having to do R&D on a completely new component seems like it would give Gemini the obvious advantage. I feel like there must be more that I'm not seeing.

I'm not sure of the difference. However I've found that the autosave has already saved results for the next season so don't depend on it if you have a mission go awry :(. Saving before you hit end season though in a normal save spot seems to avoid this.

I've found that I'm perhaps to cautious I can't seem to beat the Russians :(, but I wait until my components have nearly 90% reliability before launching, obviously NASA took risks I didn't :)....

The more you use the same components the less they seem to fail.

take the risks around 80% do some unman tests as well you can jump as well you don't have to do every step

You could try the EOR approach. You don't even need a Saturn V.

I'd never heard of this idea until playing the game and it looks crazy. A shame it was never tried in real life.

That is a very good question (the initial one). I have won the moon race by using Gemini DA, just because I already had a verified, 100% reliable component.

But I was wondering - why in the real life did NASA decide to go with Apollo? Does anyone (anyone here, obviously. There are people who made that decision, and a simple "yes" answer is not what I am looking for ;) ) know why?

i think even the Saturn V could not handel the weight you may have to look it up

According to NASA sources it was simply a matter of the available techs and resources, which made it more feasible for LOR to achieve JFK's deadline, than direct ascent or EOR.

The decision between direct ascent or some kind of rendezvous was made quite early:
https://history.nasa.gov/SP-4203/ch2-4.htm

Now when talking of the decision between EOR and LOR is when things get interesting.

Note the lack of any justification for the dismissal of the EOR concept in this NASA page, which would probably have meant establishing a space station and might have had an even greater effect on technology development (also according to this page):

https://www.nasa.gov/centers/langley/news/factsheets/Rendezvous.html

This paper sounds very true to the internal dealings of a government organization in the 60s, and the personality of von Braun, which was ultimately (according to many historians) the one that settled the argument with his endorsement of the LOR concept over EOR:

https://repository.si.edu/bitstream/handle/10088/29808/Neufeld--WvB%20and%20LOR.pdf