Last year I was looking at fares from London to Tel Aviv. As a US resident, I started with Web sites aimed at the US market; later I checked sites aimed at European markets. Fares quoted on the former tended to be way higher than the latter, especially for LH I recall. I ended up booking via a British site scoring what I considered to be a very reasonable fare. My understanding is that non-European residents can have trouble booking via some European sites either because of address or because of where their credit card is issued. However, my experience confirms that it can be done in some cases.

Here's another, related tip. Once I tried to book a hotel room in London using priceline.com. It rejected my attempts to bid (not the bid itself) with some incomprehensible error message. I gave up and tried priceline.co.uk. It worked without a hitch (and accepted my bid). This was not a case where any money was to be saved but it illustrates that there are sometimes options one might not immediately think of.