When you need position (and/or attitude) data at a high sample rate, the usual technique is to combine inertial measurement (e.g., MEMS accelerometers and rate gyros), which give you relative movement data at high sample rates, with GPS information that comes at a relatively slow rate (e.g., 1 - 10 Hz).
This information is combined in a system model (e.g., Kalman filter) in such a way that the absolute position data from GPS corrects for the long-term drift components of the inertial measurements. The result is an output that has the accuracy of the GPS, but the high sample rate of the inertial measurements.
Note that the GPS output message includes a timestamp in addition to the position values. The position values give the absolute position of the receiver at the time given by the timestamp. Even if there is a lag in terms of computing that position or transferring that message over a serial link or whatever, the data within the message is always self-consistent, within the accuracy limits of the receiver.
A lot of projects (especially those done by hobbyists) ignore the timestamp and the transmission delay and simply take the position values directly as the "most current" position, but if you really care about high sample rates and low-latency data, this is an overly-simplistic approach. A properly-constructed hybrid GPS+inertial system can provide data that's "current" within a few milliseconds.
The common term for such a system is "AHRS", which stands for Attitude and Heading Reference System, and it's a common element in pretty much any autonomous vehicle control system.
Before you ask a question, it would be nice if you performed even a small amount of research. It shows a certain minimum of respect, you know? Googling "gps accuracy" will produce this government document and page 22 will give you a good idea of what you can expect with a good receiver.
With that said, what your gps unit is giving you is resolution, not accuracy. The link suggests that you should expect better than 4 meters accuracy more than 95% of the time, although there will be small number of readings with worse accuracy.
If you take a number of readings at the same location, spread out over several hours, you will see the reported location change as the positions of the gps satellites changes. So the first reading is not (necessarily) the most accurate. Separate readings are not necessarily related to each other, since the locations of the available satellites changes with time. What is related is the accuracy of all gps receivers which are relatively close to each other. This allows for augmentation. A gps receiver at a precisely known location can determine the gps location error (at that specific time) and broadcast this error to any other receivers in the area. These other receivers can use this error to compensate their own readings, and produce accuracies in the cm range.
Read the link.
Best Answer
The L1/L5 frequency pair can be used to compensate for ionospheric delay much in the same way as L1/L2. The wider spacing of L1/L5 as compared to L1/L2 will not enhance accuracy significantly.
In theory, if using all of L1/L2/L5, second order ionospheric effects can be eliminated (see here). We will see if this gives superior correction when manufacturers adopt this (or a similar) method (I'm sceptic).
The quadrature part of L5 is used as a pilot and does not carry data modulation. A convolutional coder expands the net data rate of 50bit/s into 100 symbols/s (to shape the power spectral density of the signal in space). These are phase shift modulated onto the inphase part of L5. Net rate is 50bit/s.
WAAS and EGNOS send 250 net bits/s in 500 symbols/sec. The relevant documentation (MOPS DO-229) is behind paywalls, I cannot link it here.
BTW: The term "differential" does not apply here, it is used when signals from at least two different antennas contribute to the solution (either independent receivers or antenna arrays).