So if it costs nothing to be a "sole trader", then that looks good.
The cost is you don't get the benefit of being an employee: no sick pay, no paid holidays, cost of providing your own tools, etc. Plus you get to be personally sued if you screw up (and sometimes even if you don't), potentially for millions.
The attraction of the limited company is that when it goes tits up, gets sued for millions or whatever, it's the company that is the target rather than you personally. With a few exceptions you can just walk away and be done (but, note, that anything the company buys is the property of the company, not you).
Accounts have to be done every year and unless you are supremely confident about money then you want an accountant to do it. What they cost is between you and them, but figure on around £500-£1000. If the company has done nothing all year, you still pay the accountant to do the filing. They may do it for a reduced rate but, again, that's between you and them. Figure on it being a fixed cost per year regardless.
Finally, if you're running a limited company (or as a sole trader) and you get to be an employee elsewhere, that's not a problem. You're just an employee of two places and pay appropriate tax at each place. You can keep the limited company going or not - depends on how much hassle it is to shut it down and whether you might want to use it again. You could get your limited company to sack you so you are no longer an employee of it, but you'd remain a director.
Regarding the question of why some place might prefer you to be a limited company, it's possible that it's perceived as being more business-like than some fly by night chancer. But, most likely and as someone noted previously, it is probably a VAT thing - they might assume that a limited company would of course be VAT registered.
VAT registration is useful for getting stuff apparently cheap, but note that it is the property of the company and there are hassles involved (like filing your VAT returns every month, even if you don't buy or sell anything), and not using company property for private stuff (so you might be able to justify that 60" TV on the grounds of needing a big monitor, but you shouldn't use it to watch Netflix with your cocoa before bedtime without either the company charging you for that (plus VAT) or you noting it on your self assessment).