Electronic – Smallest lithium battery backup

Like David says supercaps leak their charge to a certain extent, which is mainly a problem over longer periods of time. Let's make the required calculations ignoring the leakage.
The voltage drop across a capacitor at a constant current is given by

\$ \Delta V = \dfrac{I \cdot \Delta T}{C} \$

or rearranging for time:

\$ \Delta T = \dfrac{C \cdot \Delta V}{I} \$

\$V_{BAT}\$ is typically 3V, but for the given supercap is maximum 2.6V. Minimum for the RTC is 2V, so the allowable voltage drop is 0.6V. Filling in the other numbers this gives

\$ \Delta T = \dfrac{0.015F \cdot 0.6V}{500 nA} = 18000 s = \mbox {5 hours} \$

which isn't very long, but then you also chose a rather small supercap. A 1F/3V cap would increase your time to 23 days, but there we would have to take the cap's leakage into account, so in practice this may be around a week to a fortnight.

edit
Just picking the right RTC and supercap will dramatically improve longevity. The PCF2123 RTC can operate down to 1.1 V, and a PAS311HR supercap not only has a higher capacitance of 30 mF, but can also operate at 3.3 V. Then the equation becomes

\$ \Delta T = \dfrac{0.030F \cdot 2.2V}{110 nA} = 18000 s = \mbox {167 hours} \$

or just short of one week. A 1F/3.3V cap would be good for 7 months, or probably 2 to 3 months taking self-discharge into account.

Both do the same, the LiPo Rider Pro just has everything on board. The 1n4001 diode used in the second instructable is used to prevent battery discharging though the solar panel at night, prevents reverse current through the solar panel when the usb charging (powering the lipo charger through usb) is used, and prevents the solar panel from affecting the charger when both light and usb charging is used. The LiPo rider has the same type of protection.