Mazal Tov, Lee.
The primary distinction between their psak is as follows: Rav Mordechai Eliyahu holds that your wife must count 5 days starting from the day she saw her period before making hefsek tahara. Rav Ovadia Yosef holds that your wife must count 4 days starting from the last day she had martial relations before making hefsek tahara. Most women bleed for 5 days anyway, so this is unlikely to make a practical difference in your life (though you may be able to take advantage of this leniency occasionally).
This generally applies regardless of the type of stain or bleeding that caused your wife to become tamei. Rav Ovadia has some leniencies that follow from the way he says to count, but you're also unlikely to be in a position to take advantage of those leniencies. (And if you are in such a position, your wife may nevertheless not want the pressure of doing bedikot right away.)
Some other examples of differences in their psak are in harchakot (the rules of separating husband and wife when she's not tahor):
- Rav Ovadia permits you to listen to your wife's singing. Rav Mordechai Eliyahu does not.
- Rav Ovadia permits you to go on pleasure trips with your wife. Rav Mordechai Eliyahu does not.
- Rav Ovadia permits you to sit on a bench, couch, or swing with your wife. Rav Mordechai Eliyahu does not.
There are plenty of other differences in other aspects of niddah as well, but I don't know those so well, and don't want to list the ones I do know in detail. Rav Mordechai Eliyahu tends to rule more similarly to Ashkenazi poskim (following the Rema in many more places) than Rav Ovadia does.
If you're learning to prepare for your wedding, you need to ask your rav whom you should be holding by. I personally follow Rav Ovadia in everything (niddah, and other stuff, l'chumra and l'kula), because my Rav in Chicago is a talmid of Rav Ovadia.
If you're learning according to Rav Mordechai Eliyahu, the sefer to learn is Darchei Tahara (as Robert explained in his answer).
If you're learning according to Rav Ovadia Yosef, the sefer to learn is Torat HaTahara by his son, Rav David Yosef (it's in Hebrew, get the edition with the red cover). There's an English book according to Rav Ovadia's psak by another son, Rav Avraham Yosef named "Taharat Yosef," but I don't recommend learning from it. R' Avraham's coverage of the number of days your wife needs to count is so unclear as to be misleading.
If you're learning according to Rav Ovadia, you should also read Rav Mordechai Eliyahu's book, because Torat HaTahara doesn't have very good coverage of the calculations that you uses to know when to separate from your wife in anticipation of her period (vestot). Torat HaTahara also doesn't discuss conduct in the bedroom with your wife (nor have I found this in any of Rav Ovadia's other writings), while Rav Mordechai Eliyahu's book does. Both of these are things you'll want to know.
Regarding the former omission, my rav told me that Rav Ovadia wrote about vestot for his sefer Taharat HaBayit (on which R' David's book is based), but the chapters were lost before they could be published, and Rav Ovadia never rewrote them.
If you have the time, and are interested in a deeper understanding of all of the concepts and issues in hilchot niddah, I strongly recommend The Laws and Concepts of Niddah by Rabbi Zvi Sobolofsky, a Rosh Yeshiva at Yeshiva University. I've only skimmed that sefer myself, but here is a review. (But don't read that as a substitute for the Sephardic rav who you'll be holding by. It's more important to know the halacha l'ma'aseh than it is to know the concepts.)
