If both your laptop, and desktop are using the same internet connection then they will have the same IP. It is hard to tell what is a flagged IP tbh, as jagex does not say when you get banned, its for a flagged ip XD But, if the last 2 numbers are changed, yes in general it wouldnt be flagged anymore. But if its a huge probably for jagex this ip range.. it is possible if a proxy company (for example) sells a lot of ip's and see's a pattern that only the last digits are changed, they know they are all from the same place because they are all on the same ip range so that server will be flagged. This is why you need to be concerned about who you buy proxies from, and sometimes people buying from companies heavily realted to RS specificially the ip's can be flagged even if never used before or for botting.
I know I went a little off-topic, but I hope I explained some ip related information
Free proxies/vpn's are not worth your time as they will be flagged 99.99999999% of the time, so it just wastes your time =/