Many reasons including different time zones around the world, scheduling difficulties, promises to broadcast on one station first, to generate word-of-mouth interest.
**There are a few reasons for this.
Firstly, at this Metro article discusses, British shows are often scheduled at fairly late notice whilst American shows have their timeslots booked months in advance. This is one of the contributing factors explaining why the US gets UK shows so much later (sometimes months later). Now, this doesn't answer your question about why the US shows are later in the UK, but it is a useful bit of information to explain one of the issues in syndication.
Secondly, an Entertainment Weekly article featuring an interview with a PBS executive, when asked about why Downton Abbey aired in America months after the UK air date, featured this section:
I think as we have looked at this whole issue of spoilers and thought
about how best to steward the property and also think about the
viewership, we considered a number of factors in the scheduling of
Downton,” [PBS president Paula Kerger] said during a panel at the Television Critics
Association press tour, which concludes this week in Los Angeles.
Those include, she said, attempting to avoid clashing with the slew of
network premieres in the fall and harnessing the opportunities for
word-of-mouth promotion. “The fact that word-of-mouth travels after it
airs in the U.K. has actually benefited us … we kind of don’t want to
mess with that if it’s working so well.”
In other words, because the show aired in the UK and was popular, word of mouth spread across the pond and increased anticipation in the US, which they felt improved their ratings. It's an interesting argument, and its accuracy is of course debatable - but it's what they feel.
Thirdly, channel executives can be apprehensive about airing shows that compete with "traditional" favourites. For example, Friday the 22nd November in the UK was dominated by I'm a Celebrity, Get Me Out of Here, with a 34.5% market share (note Digital Spy have daily ratings on their site). In other words, a third of all TV sets in the UK on that night, at the time, were watching I'm a Celebrity. If a US show at the same time were to be broadcast in the UK, it would be competing against it and thus would be likely to lose. This is a consideration. Note, I'm aware Friday is a bad example due to the graveyard slot in America, but roll with it!
Fourthly, often deals are made restricting where shows are broadcast first. From an article in The Atlantic, Eddie Schmidt, an Oscar-nominated filmmaker, said:
If you're getting money from X company they may have a foreign service
deal with Y company. For example, if HBO funds your film then you are
of course beholden to having it released on HBO and having it roll out
on other platforms as they permit." Schmidt continued, "If you're a
large multi-national, or a US company that has made foreign 'output
deals' for your film or shows in foreign markets—this could be
TV/broadcast, but could be theatrical too—then you can only do what
they permit; there can be 'holdbacks' on digital exposure. They want
exclusive hold of the product for X amount of time."
In the same article, an ITV spokesperson anonymously said:
"We're looking to build a strong international content business by
creating our own content, achieving strong ratings on ITV in the UK
and then selling it around the world." Commercial networks fear that
if a show is available simultaneously and/or before it airs on TV that
it would cannibalize the audience. Producers fear that if they do a
multi-platform release it may hurt sales for TV rights.
So in these cases, the broadcasters are either financially responsible for broadcasting on one platform first or simply feel it's in their best interests to.
Fifthly, time differences can come into effect. If a show is broadcast in the US at 9pm, then the equivalent "live" broadcast time in the UK is 2am. That's a heck of a difference and it might be felt that this wouldn't be worthwhile. Why chase a program and pay extra to get a "at the same time" broadcast if the showing will be at a crazy early hour - when you could pay less, broadcast it the next day and get more viewers.
Finally, whilst piracy IS an issue, not everyone feels it's as "big" an issue as you suggest. From the Game of Thrones Wiki:
Observers, including series director David Petrarca and Time
Warner CEO Jeff Bewkes said illegal downloads did not hurt the series'
prospects, as it benefited from the resulting "buzz" and social
commentary, while the high rates of piracy did not significantly
translate to lost subscriptions.
Obviously, they could be lying to save face, or simply avoid discussing it in detail, but their views are worth considering.
However, having said all that...
Broadcasting shows simultaneously is definitely happening more and more. For example, Game of Thrones was shown at the same time in the UK and US. You already highlighted one of the reasons why - piracy. Game of Thrones was the most pirated series in 2012, 2013 and 2014, and the move to showing it at the same time world wide was an attempt to reduce this (despite the comments pasted above).
With regards to piracy in general, Tim Gray, the editor of Variety, commented:
“It’s hard to monetize these things in the film world—but even harder
in TV,” Gray said.
“Everyone agrees it’s a crisis, but nobody quite knows what to do
about it… Netflix, Hulu and others are trying for options, but there
are no easy answers.”
So no one quite knows what the answer is, and different techniques are being trialled.
The Doctor Who 50th Anniversary episode was broadcast around the world at the same time (setting a World Record in the Guinness Book of Records). This was done to avoid plot leaks and, given the record ratings, didn't seem to affect ratings the way some of the suggestions I posted at the top of this answer postulated.
There are a number of traditional reasons listed above as to why broadcasters don't want to air shows at the same time. However, evidence is apparent that when this is done the ratings aren't harm and can actually increase. However, whilst a few shows have trialled this, it isn't commonplace - yet. There seems to be a consensus that it could be, possibly very soon, but it would require a few of the big broadcasting corporations to take a (possible) risk and do it - something they don't seem quite willing to do yet.