The major factor in broadband speed is the length of the bit of copper that carries the broadband signal - so this is between the two bits of equipment at either end (your broadband modem/router and the DSLAM at the other end). The longer the cable the slower the speeds possible.
In days of old the wire went from your broadband router/modem all the way to a box at the exchange. This could be miles, and that means broadband speeds under 10Mbps.
FTTC is BT Infinity (now called BT Fibre 1 or BT Fibre 2) means that the broadband part (which goes over the same copper wire as the phone) is terminated (i.e. the broadband signal is yanked off the wire) at the cabinet (FTTC = Fibre to the Cabinet). This shortens the copper run significantly for the broadband signal (now often a few hundreds meters instead of a mile+) and broadband speeds are up to ~80Mbps. But BT had to move the broadband termination equipment (DSLAMs, etc) to the cabinet, which means the cabinets now needed to be powered (they didn't before), plus they needed to have fibre installed between the cabinet and the exchange to carry everyone's data, so lots of faff, planning permission for bigger cabinets and digging up roads.
To make the length of copper even shorter the plan is to shorten it to zero by bringing FTTP (fibre to the premises). This is the BT Full Fibre option. Speeds up to ~1Gbps. No "broadband" in traditional terms at all as your local Ethernet network will go out of the router on the fibre. You'll need an engineer to come and yank some fibre optic cable up alongside your existing copper pair(s) and then drill it through the wall as it can't be done over the existing copper cable. Fun.
Your phone signal still travels over the existing copper wire all the way to the exchange, at which point it is digitised by an 5ESS or equivalent.