>>Why will the current be zero when it hits 13.75v?
When the battery's internal voltage hits 13.75V there will be no differential between the charger and the battery, so no current. If you check the unloaded float voltage of that lead acid battery, fully charged and disconnected from anything, I believe you will find it's over 13V. 13.75 is an excellent choice for the charge point voltage.
>>What if it hits 13.75v before the battery is done charging?
It's normal to see the charger at full voltage across a battery that hasn't REALLY finished charging. To do that, you need to measure the current.
>>Is that due to Rs?
>>Also, what wattage resistors should I use?
Rs just makes the LM317 appear to have a 2.2ohm power resistor in series with it. Incredibly, with a maximum output of .68A, all those resistors can be quarter-watt. But use a 1 watt for Rs, a really drained battery might just pull more more current.
>>So could I just leave out Rs then since it doesn't need any current regulation?
Probably. The battery can easily handle more current, Rs is mainly in there to reduce the load on the LM317 when it's dealing with a heavily discharged battery. It DOES have some internal protection against over temperature (high wattage). Help it out and give it a good heatsink, this is a series-pass regulator. The wallwart will almost certainly put out more current than 680ma if the voltage is loaded to under 12V. Try hooking an ammeter (on a high setting) in series with the wallwart and a heavy load (a headlight or something). See how much it can REALLY put out. THat's what the regulator is going to have to deal with