given a choice between the 2, I would choose win 8.1.
better memory management, faster start up and shut downs, newer OS meaning still supported, little to none of the win10 phone home stuff, generally faster than 7, far more secure than 7 and by using Classic Shell it looks a lot like 7.
Can you demonstrate these points in real-world terms?
I see pretty similar memory usage scenarios on Win7, 8x and 10.
While Win8x and 10 play a few games to ensure that it gets past the 'loading Windows' screen quicker, IMO the trade-off comes in later by loading more stuff later, so while the user can technically start using the machine sooner, it doesn't mean it's any quicker. Furthermore, switching off the screen before the computer has finished shutting down is just plain cheating, and I fail to see the benefit in chopping off 3 seconds and losing the F8 startup menu in direct exchange.
"Newer OS meaning still supported"? Win8x isn't going to get any meaningful functionality updates. While technically it may still be in the support phase that entitles it to functionality improvements, it won't get any; just the fact that it's had nothing since Win81 U1 and Win10 only has DX12 support is pretty telling. The only valid argument with regard to support IMO is Win7 is supported with security updates until 2020, Win8.1 until 2023. Whether in reality that will mean that Win81 won't be dropped prematurely by third parties like Vista was because its user base is just as small, I don't know.
"Generally faster than 7", my question speaks for itself.
"Far more secure than 7", my suggestion here for you is to go through the patch list for say a 6 month period and note how many Windows vulnerabilities applied to all 'current' versions of Windows versus what you regard as "less secure" versions of Windows only. While newer versions of Windows have some under-the-hood improvements, if those don't translate into vulnerability mitigation or preferably immunity, it doesn't mean a great deal, does it?
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To answer the OP's question directly: Go with Win7, IMO. Your computer is very low spec and pretty ancient, ASUS doesn't list drivers for anything newer than Win7. For example, I have a laptop that I picked up second-hand that originally came with Vista but lists Win7 support. It runs Win7 fine, but I tried Win8.1 on it and the best Intel graphics driver doesn't render as well as it does on Win7, so for example something as basic as text manipulation suffers from graphics corruptions. Bluetooth support also went bye-bye because the Dell combo adapter doesn't work 100% with Win8.1.
Generally I'd argue using a newer version of Windows over an older one, my main reason being support (Win7: 2020, Win8.1: 2023, Win10: "forever, last version of Windows"). However, Low spec PCs and Win10 are not a good mix because of the feature updates and the processing time attached to them. Even a Sandy i5 HDD laptop I have in took several hours to process 1709 before wanting to finish installing with during a reboot. The OP's netbook would probably take 10 hours or more. If the computer was say that Sandy i5 I laptop mentioned and I could just pick any version of Windows without licence worries, I'd go with Win8.1 (though frankly unless you have random Windows licences lying around, I don't see how you could legitimately have that choice) for the longer security patch support, not for any other reason. However, I can see a distinct possibility that third party software makers will betray Win8.1 users who choose for this reason, if anything by supporting Win7 for as long as they do Win81 (which is what happened with XP and Vista).
OP, if you have the means, stick an SSD into that netbook. I certainly wouldn't pay for a newer version of Windows or bother pirating. IMO an SSD is the difference between a computer that you will forever be grinding your teeth while waiting for it to finish the simplest of jobs, and one that is bearable. Of course, user preceptions and expectations of performance vary, but that's my two cents. I've never encountered a customer who thought that their early-generation Atom-powered netbook performed in a way that they appreciated.
Doubling the memory would be a good idea as well (to 2GB).