What is the difference between the closed system boiler, and the conventional boiler?
Traditionally, as I believe you have, the primary system (ie the water that goes round the rads and the heating coil in the hot water cylinder) is "open vented". Up in the loft is a small feed and expansion tank - as the water in the system warms up, it expands, and the level in your F&E tank will rise a little. As the system cools and the water contracts, the level drops a little. There's a ballcock for filling the system and maintaining the level - but normally this will stay closed pretty well all the time. There is a small pipe from the bottom of this F&E tank to the system and the water flows through this as required.
In addition, there is a vent pipe which comes from a high point of the system, and any trapped air can vent up through this pipe - it's normally turned over so that if any water goes up it and it "pumps over" (shouldn't happen in a properly designed system) then it purs back into the F&E tank.
These days it's more common to have an unvented primary. There's no F&E tank, instead the system is filled from the cold mains via a filling loop (which includes more than one non-return valve). To allow for water expansion there is an expansion vessel which has a flexible diaphragm to separate the water from a pocket of compressed air.
Both systems have pros and cons, and I'm sure that just mentioning that will start a debate on why and open vented system isn't appropriate for any installation
Now, back to your situation.
There is another option I suggest you at least consider - and don't be put off because some people here will tell you that they cannot ever be a sensible option. That is a thermal store (TS).
What you could do is replace your current hot water cylinder with a thermal store, heated by your current boiler. The simplest installation would mean leaving everything as is except you'd disconnect the cold water storage cistern in the attic, and connect the mains cold feed directly to the cold inlet on the TS. You'd also connect anything else (possibly the cold tap on the bath) that runs from the cold water cistern to the mains. Now all your hot water and cold water taps etc run from mains pressure -
but unlike an unvented hot water tank (eg the Megaflow) you do not have a pressurised tank, so the issues around that do not occur.
For a little more plumbing work, you also alter the central heating so it runs from the store (and so only indirectly from the boiler) with it's own pump. Because the flow rate through the heating doesn't affect the flow rate through the boiler, the pump can now be a modulating type and give you a very quiet central heating system.
So, how does this miracle work ?
You have a tank of hot water - this can be the same water that goes round the radiators and boiler (there are several options). Winding it's way up inside the tank is a tube through which your hot tap water flows - and it absorbs heat through the walls of the tube so it comes out hot at the top.
A different variant, usually called a heat bank, has an external plate heat exchanger (and another pump, and flow switch). These have their own pros and cons.
Cost wise, a TS will probably set you back a bit more than an unvented cylinder, but the advantages may be worth it - that's something only you could decide.
EDIT: One disadvantage both a TS and heat bank share with a combi and unvented cylinder is that they rely on you having an adequate dynamic flow and pressure from your cold mains.
But you can have an immersion heater and so have a backup if your boiler breaks down - this may even be able to contribute to heating if you plumb things up that way.
So, over to the usual guys, let the TS bashing commence