How Is the Magic Core Set 2015 "Deck Builder's Toolkit" Different From Other Core Set Offerings With Respect To Cards?
It depends on what cards you have and what you are interested in becoming.

If you are a beginning Magic player (i.e., you have no cards at all) then I would suggest getting the Deck Builder's Toolkit. This includes plenty of cards from all 5 colors, and you can easily craft a (fairly weak) deck from what you get out of it. 

It should also come with 4 booster packs, which is nice, and the 100 lands, which you will really need. However, you will probably only get 4 rares out of the kit (the 4 rares from the booster pack).

If you at least have a collection of Magic cards, and want to get powerful rares, I would suggest getting a fat pack. They include 9 booster packs, so you are guaranteed at least 9 rares, though you have a good chance at getting foil rares or mythic rares (or both!). You will also get 80 lands, which is nice. 


  Does the toolkit have inferior cards compared to the fat pack?


The toolkit will come with 5 rares in the fixed card pool, plus 4-5 rares from the booster packs &mdash; and you will get a good amount of decent uncommons and commons to work with. The fat pack will come with at least 9 rares with a high chance of pulling a mythic rare, and will also include 27 uncommons.


  If I were to compete with just cards from the deck builder's toolkit, would I be at a horrible disadvantage?


It all depends on who you are playing with. Just playing kitchen table Magic, you should be fine if you play with people who buy basic decks. However, at a tournament, you will get eaten alive by pretty much everyone. (Even some casual players with decent decks will destroy you.)

Also note, buying a fat pack is not really a good way to start, either, because although you will get those powerful cards you need in a deck, you won't get the "backbone" that your deck needs to survive. For example, you may get a powerful late game card from the fat pack, but if you don't have the cards to stay alive that long, that card is pretty much worthless in your deck. The Deck Builder's Toolkit has cards to help you in that circumstance.

In conclusion: it all depends on your circumstance. If you have no cards at the moment, buy a Deck Builder's Toolkit. If you have a good base of cards to work with, buy a Fat Pack. (If you have the money, I would suggest buying both!)