The clocks are the same, they just look different. I have the blue one just like yours. I had as seriously bad time getting any joy from it. This was partly due to my being a newbie, but far from wholly. I believe there is a swag of confusing and out-of-date junk out there - and badly written too. Since you ask the question, I assume you two are newbies. Being a newbie puts you at the disadvantage of having the latest gear, thereby having to sort the wheat from the chaff, and quite a lot of that chaff is in the Arduino playground.
About the only decent up-to-date tutorial written for the DS1307 is here
http://bildr.org/2011/03/ds1307-arduino/ Read the first line in the code twice, indeed they write it twice.
You only need four wires. Most of the terminals on the blue board are duplicates and it doesn't matter which one you use.
There are two sketches, one for setting the clock and the other for running. You write the setting time in the first sketch and the start point is when you release the reset button. Needless to say, the two sketches can be merged if you wish, and it might be an interesting programming exercise involving serial read, but I haven't bothered. My module came running on Shanghai Time.
