alt Hacker NewsFor changing browsers, IPs, user agents, etc., it probably rounds to 0%. There are some edge cases booking flights directly from national carriers from within/outside the country. For using third party OTAs for hotels, I'd guess maybe 5% or so on average depending on your usage, but the financial tradeoff isn't directly comparable. You can probably save a bit by booking third party, especially if you haven't bothered signing up for hotel loyalty programs [1]. There are...let's say "discontinuous" long tail issues with booking hotels third party, where probably 95-99% of the time you'll have no problems, but when there is one, you'll have a much better chance of having a very bad time. The price gap on flights is probably considerably smaller (commissions on flights are much lower than commissions on hotels), but if you have any sort of problem (e.g. a flight being delayed when you have a layover), you have a much better chance of getting screwed.
Personally I mostly use Google Flights for flight search (or seats.aero for award travel search) and booking.com for hotel search, and then book directly. I've booked probably 100+ hotels through booking.com and then quit entirely after hitting that long tail bad experience.
[1]: I'm a few years out of date on this and don't know the current legalities, but major OTAs (Booking.com and Expedia, who control probably 80-90% of third party hotel bookings) used to have a "most favored nation" clause that prevented hotels from advertising lower prices through other channels without a previous commercial relationship. Signing up for a loyalty program qualifies, so being logged in for direct hotel bookings can sometimes get you cheaper room rates. (In the US, things like AARP and AAA memberships also work.) I know the EU was pushing back on these clauses but I'm not clear on what the current state of things is.
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There are a few more modern OTAs that rebate a portion of the OTA commission back to you (see Rove, often rebating 15+%) and more that rebate a flat rate (Chase at 8%, Amex at 5%, Cap1 at 10%, etc). The "member pricing" is often 1-2% lower at most, so going through an OTA should get you around 10% off rack rate at minimum, more if commissions are particularly high (hotels pay about 20-30+%, so it's not uncommon to see 20% returns on Rove. Highest I've seen is 50%!).
OTAs can play "member" pricing games too I think so there's even better deals to be had.
So you definitely need to comparison shop, I make my agent take the pricing from all the OTAs and calculate a post-rebate price, then pick the lowest one.
Booking and Expedia and other marketplaces also have a habit of packaging weird short-term rental operations to seem like a hotel. I wound up in a strange city where there was no reception, no lobby, and I later figured out I had failed to confirm my room by not replying to an email that a vendor sent me which I never saw.