A very quick glance at the specifications of these two compact pocket cameras shows that they are almost identical. Here are some of the similarities:
Size: 3.4" wide by 2.2" high by 0.7" deep
LCD: 2.5" with 230,000 pixel resolution
Zoom: 3x with very similar focal equivalents
Image Quality: both are excellent for compact pocket cameras
Aperture: not exactly equal but not enough difference to make a case one way or the other
ISO range: auto, or 80-1600
White Balance selection: 6 positions and manual preset
Image Processor: Canon Digic 4
The differences are only in the Megapixels and the video capability. The Canon PowerShot SD780 IS has a whopping 12 megapixel image sensor, while the Canon PowerShot SD1200 IS has only a 10 megapixel image sensor. If you are a megapixel watcher, this may be enough for you to go for the SD780 IS, but it really is not enough of a difference to base your final decision on.
The biggest difference is in the video capture feature. The Canon PowerShot SD1200 IS has decent video with capacity of capturing 640x480 video at 30fps (frames per second), which is similar to previous pocket cameras. However, the Canon PowerShot SD780 IS has the capability to capture HD video at 1280x720 pixels and 30 fps. This is the same video capability and format as some much higher end cameras.
So, if you want to be able to show your pocket camera video on your HD TV with excellent quality, the Canon PowerShot SD780 IS is the camera for you. The additional is well worth the upgrade in video.
If video is not your thing, then by all means, save the and get the Canon PowerShot SD1200 IS. The image quality for still photos is pretty much identical. Use the money you save to get a memory card for your new family pocket camera. (By the way, you will have to spring for the memory card with the SD780 too, in order to record that great video.)