Have to clear up what seems to be a common confusion about LDO regs.
A 5V LDO and an 7805 will drop the exact same voltage
when supplying the same current with the same input voltage above (U_out + U_drop) (The laws of thermodynamics still applies I believe
The only real feature of the LDO compared to a normal regulator is simply the minimum drop out voltage, which for an 7805 is minimum 2.5V (usually around 3V though), while the LDO can be anything down to 100 mV or less (depending on output current).
LDOs exists in different current and power capacities.
The higher current types usually have a smaller drop on a lower current (a 7.5A regulator supplying 1A can have a drop of eg. 1/3 of what a 2A regulator supplying 1A have).
An 7805 will go out of regulation when its input goes below around 8V, while a 5V LDO wil go down to at least 5.6 before loosing regulation - a big difference when using a 9V battery.
A 9V battery contains 6 cells. A single alkaline primary cell is up to around 1.65V for a short time when new and is considered flat at 0.9V.
That means that a new 9V battery is up to slightly below 10V while it's considered flat at 5.4V.
An 7805 will only use a minor part of the capacity of a 9V battery, while an LDO will use them up completely without loosing regulation.