What do you gain from MEng over BEng? Well, a year of further study is a big one. You'll do more specialised things in the fourth year and this may help your career prospects. On some graduate schemes I've seen, graduates with a Masters may get a £2k bonus to their salary (£25k up to £27k, typically).
Why MEng over MSc? Well, an MEng is an undergraduate degree covering Masters content, but it's still one degree. The good news is that this means it's covered by student finance. An MSc is a postgraduate degree (both the MSc and MEng are typically one year long) and is not covered by student finance, but it can be done any time after graduation and gives you a second degree in your title. An MEng must be completed in the same area as the BEng (as it's an extension of it) and must be done in basically one go. I don't think you can leave and come back to do the MEng. The MSc lets you change engineering field so you can get more experience in other areas.
The MEng or MSc will both allow you to become chartered. You can become chartered with a BEng, but this requires further training. A friend of mine at JCB told me he is on a training scheme with them to get up to chartered standard on his BEng. Being chartered is a way of showing professional and academic competence. You need experience on engineering projects to become chartered, so employers can quickly see that you're a suitable candidate. It can also give you networking with other engineers through your institution.
You typically choose to do your MEng in the second year, albeit probably late in the year. As mentioned by a10, you'll need to be academically suitable for it. This is why MEng entry requirements are slightly higher, but it shouldn't discourage you from trying it out. Some universities may tailor the degree differently in the third year for those going onto the MEng programme. Quite often though, the first year is a standard year for all people on that course. BEng and MEng students will probably share the first and second year and then start to branch out from there. Sometimes it is purely a fourth year with no other real difference.
Funding should come under student finance for the MEng as it's still a Bachelors degree. It will just be 4 years of funding instead of 3. They will manage that for you though. Personally, my goal is to finish my BEng Mechanical (with a placement year) and then use the money from the placement year to fund my MSc Aerospace. It would be well worth considering taking a placement year as this will give you a massive advantage when it comes to finding a job afterwards. Unfortunately, this does mean it will take me 5 years to get my degrees and be done with it, but I think it's wise in the long run.