I suspect that the Photon is faster, but I don't know of any official tests. Connecting to wifi seems to vary a little bit every time depending on distance, obstructions, planetary alignment, the Avengers, etc. Wifi on my home network is usually pretty quick to connect for any device, but every now and then it will take several seconds for any device (Core or even a fancy laptop). Unless it is taking a long time, I doubt the time it takes to connect will drain much from your battery. In most of my testing, the Core takes 15-25 seconds to connect to wifi and handshake with the cloud.
You may be interested in the Spark Battery Life Testing thread I posted a while back where I tested the Spark Core on lipo batteries with various sleep cycle durations. The sample code for those tests is in the paragraph below the picture. Your mileage may vary if you need to keep the Core/Photon awake to watch for gate openings, so the SYSTEM_MODE(SEMI_AUTOMATIC) that @ScruffR mentioned above may be your best bet.