I was unaware that HP had a store here in the UK so I've had a look at the link you sent regarding this model
HP 17-x013na Laptop
Why is the above so different (aside of the price) any different from this hp 15-ay069sa
I am looking at the specifics but really understand none of it!
I'll try to keep this simple, but bear with me - CPUs are extremely complicated. I don't claim to understand anywhere near the whole picture.
The short answer: Because of the CPU. The Intel Pentium N3710 is based on Intel's Atom architecture, which is designed with two goals in mind: to be as cheap as possible while consuming as little power as possible. Atom chips first showed up during the "netbook" craze nearly a decade ago, when everyone and their dog bought a cheap 7-10-11" "laptop". While this is significantly faster than 10-year-old netbooks, and it's on the higher end of the Atom chips made (denoted by it being named Pentium and not Celeron or Atom), it's by no means fast. It's a processor made for a
passable web browsing and text editing experience, and not much more.
The Intel Core i3-5005u in the 17" is an entirely different beast. It's based on Intel's Core architecture, which scales in performance from "ultrabooks" (thin-and-light premium laptops in the ~13" size segment) to workstation and server chips. The i3-5005 is on the low end of this scale (the u on the end of the name denotes low power, i.e. ~15W max, thus suited for ultrabooks and other thin laptops with decent battery life), yet even at that end of the scale it vastly outperforms the Atom line.
The Core line are often spoken of as "big" CPU cores, i.e. cores meant for high performance (big also reflects their actual size when compared to "small" cores), while the Atom line are "small" cores (typically used for "ok" performance, low cost and battery savings). The size difference is rather dramatic, which reflects on how two "big" cores can outperform four "small" cores.
Another factor is power: the 5005u is a 15W chip, while the N3710 is a 6W chip. That doesn't denote constant power usage, but maximum continuous usage. As a basic rule, higher clock speeds and bigger cores consume more power. The 5005u runs at 2GHz, while the N3710 runs at 1.6 - but it can boost up to 2.6GHz when needed. This "speed boost" is counteracted by power limits - the chip can't continuously consume more than 6W of power, and if that happens it has to stop boosting. The higher power limit of the 5005u allows it to run at 2GHz at all times - but it also consumes more power, and given the same battery size would then lead to far shorter battery life. And again, due to smaller, less powerful cores, the Atom chips - even with four cores at 2.6GHz - would perform worse than two Core cores at 2GHz.
An example of their performance: In
PassMark's PerformanceTest, the N3710 scores 1870 (that's the average of the 81 people that have run that benchmark on PCs equipped with that CPU), while the 5005u scores 2922 (avg. of 779 users). While this is only one test, and as such not representative of all workloads you can run on a PC, it's roughly representative.
Then again, the i3-6100u in the Lenovo I linked is also significantly faster than the i3-5005u. This is due to it being a) a slightly higher end Core chip, and b) one generation newer architecture, which means some performance gains.
A very general rule of thumb: within the same product segment (i.e. non-gaming laptops) Intel's performance hierarchy goes something like this: Atom -> Celeron -> Pentium -> i3 -> i5 -> i7. This gets complicated by some lines like the "Core m" series (premium ultra-low-power, ~7W chips - as fast as an i3 or i5 in short bursts, but not in the long run). The 12" HP 2-in-1 (tablet with a keyboard dock) that I linked used a Core m3-6y30, which
is actually faster than the i3-5005u in PassMark's testing.
Hopefully this isn't too much of a wall of text. Tl;dr:
Because of the CPU.
Of those PCs, with your workload, I'd actually recommend the tablet. It would feel the fastest in day-to-day use (thanks to its fast CPU and SSD storage (yes, this affects performance, and would take another mile-long explanation)), probably has passable battery life, and is far smaller and lighter (=more portable) than any of the other candidates. If you need more storage, buy an external hard drive or high-capacity memory stick. It would be worth it, IMO. The screen and keyboard are small, which is a minus. But I'd rather live with that than the abysmal performance of the alternatives, or the weight and bulk of the 17".
Of course, the Lenovo Ideapad I linked outperforms everything here, at your target price, and is a 15" standard laptop. So there's that to consider.