The on-property vs. off-property is one of the age-old Disney conundrums, and it is actually tied-up with your second question about rental cars.
Disney hotels are almost always more expensive than Non-Disney properties with similar amenities. Sometimes, in the case of Disney's "Deluxe Resorts," the difference is significant ($100's per night), compared to a comparable 4-star resort off-property that you can get using the Hotwire or Priceline links on this site. However, for the extra cost, you do get things that off-property resorts cannot. First off, Disney resorts do not charge any "resort fees," or parking fees. In addition, if you are staying at a Disney resort, you can park for free anywhere on property, including at the theme parks, an $18/day (I think) value.
Also, if you are staying at a Disney resort, you can use their "Magical Disney Express" service (MDE), which is Disney's airport shuttle. Free of charge, they will pick you and your luggage up from the airport and take you to your resort, and at the end of your stay, do the same for the return trip to the airport. This is all free of charge, and eliminates the need for a rental car. However, if you do this, you are essentially "trapped" at Disney World for the duration of your trip -- not necessarily a bad thing, right? :) But, if you want to spend a day at Universal or eat (less expensive) meals at off-property restaurants, you'll have to find other transportation (often a taxi), which can get expensive.
Possibly the 3 best amenities that Disney resorts provide are: location, location, & location. The Disney World property is huge and sprawling, but many of its resorts are either within walking distance of one or more theme parks, or no more than a short boat or monorail ride away. Some of the less expensive resorts on property are a bus ride away, but the Disney transportation system is by far, more convenient and efficient than any shuttles run by off-property resorts. If you drive from an off-property resort, be prepared for up to an hour or more from the hotel to the park turnstyles, depending on which park you visit (Magic Kingdom is the most remote.)
Finally, Disney resorts have that "Disney Magic" -- it's hard to describe, but is a culmination of elaborate theming with attention to every detail, the pleasantness of every employee (Disney employees are actually called "Cast Members"), and the fact that you are really secluded from the "outside world." This is a game-changer for some, but will get a "Meh" from others.
Please note that there are some resorts (Swan & Dolphin come to mind, but there are others) that are located on Disney's property, but are not run by Walt Disney World. That may or may not have some of the above mentioned amenities, but may also have the feel of off-property resorts.
So, having said this, why stay off-property? Well first, there is the cost factor. You can find resorts (especially using Hotwire or Priceline) for much, much less. We have actually found some gems off-property over the years (some not-so-much-gems, too). And, depending on which park(s) you plan to visit, you can find some resorts that are actually closer than Disney resorts. For instance, some hotels in the "Maingate West" area will be closer to Animal Kingdom than most Disney resorts. Also, as you might imagine, everything from food to souvenirs will be cheaper outside of Disney.
Off property resorts usually have some kind of shuttle to the various parks. Some are free, but many charge. Schedules are often very limited. You can get by without renting a car during your stay, but it might make more sense to rent one. Finding economical transportation to & from the airport might be the deciding point in whether or not to rent a car. If you do decide to rent, make note of which resorts offer free parking, and which do not, and remember that the theme parks will also charge for parking.
One other thing we have found, is that while the Disney resorts have sometimes magnificent theming throughout the resort, the actual room quality does not always live up to the rest of the resort. We have found that the furniture, flooring, bath fixtures, etc. are often nicer in the less expensive off-property resorts than in even the Deluxe Disney resort rooms. Also, note that you can visit any Disney resort, even if you are not a guest there. So, you can still stay off-property, but still consume some of the resort's "Disney Magic." We have done this before, for example, taking the monorail or boat from the Magic Kingdom over to the Grand Floridian resort just to take a look around.
In the end, it all comes down to your budget and what is important to you. For some, the "Disney Magic" goes a long way. For others, it's not worth the extra cost. Good luck with your resort purchase, and have a great trip!