One of the principle decisions in planning for any trip is when to go, but for Walt Disney World and Disneyland, the answer to that question is hotly contested and argued. With all that in mind, I don't think there's necessarily one answer, as the "best week" to visit really depends on you and several preferences; more specifically, crowds, weather, and refurbishments. Now, let's take a look at my ideal weeks to visit both parks. Keep in mind, both basically have the same "best weeks".
Best Weeks for First-Timer Visitors
April 21 to 27, 2013
May 4 to 10, 2014
April 26 to May 2, 2015
Best Weeks for Repeat Visitors
December 8 to 14, 2013
December 7 to 13, 2014
December 6 to 12, 2015
Whether you're a repeat visitor or a first-timer, you'll want to be able to ride all of the popular rides, but during extremely slow times of the year at the resort, they're often brought down for refurbishment. This usually occurs from mid-January to late February (sometime early March) and early September to mid-November (late October to mid-November sometimes don't see quite as many refurbishments). That being said, because both parks operate all year-round, smaller attractions still may need to close during times outside of these windows. The only times where refurbishments are really unlikely are the very busy times of the year like spring break, Easter, summer, and Christmas. Generally, refurbishments are least likely in April and May as there is very little time between the end of that busy season (being the Spring).
Regardless of being a first-timer, everyone wants good weather. From mid-May through mid-October, weather is hot, humid and rainy making being in the parks exhausting. Additionally, from June 1 to November 30, you are in hurricane season and run the risk of hurricanes, though they are rare, and only generally occur between late August and early October. March through mid-May and late October through late December offer the best weather, but in November and December, temperatures can get low at times, so be aware. January and February generally seem frequently cooler temperatures. Personally, early March through mid-May provides little rainfall and humidity with ideal temperatures around 80 degrees.
Finally, crowds are at their lowest in January, February, and September. However, crowds are slightly higher (but not too much) starting about a week or so after Easter through mid-May and from October through two weeks before Christmas with the exception of Thanksgiving and the week before and after it. Crowds are at their highest during Martin Luther King Jr. Weekend, President's Day week, from early(-ish) March through Easter, Memorial Day through mid-Auugst, Thanksgiving, and Christmas/New Year's.
Keeping all that in mind, the times of year that seem to be constant in all three of those areas are a bit
after Easter through mid-May and early November through mid-December (excluding Thanksgiving and the week before and after). Now, both times of the year are excellent to visit during but what differentiates these two periods is one thing: holiday decorations. Ironically, that makes me recommend first-timer visitors traveling during that April and May period, as while all of the decorations are absolutely extraordinary, the additional planning that comes with them can become extremely overwhelming and stressful (i.e. purchasing tickets for Mickey's Very Merry Christmas Party, booking a Candlelight Processional Dining Package, etc...), while the spring provides similar (sometimes lower) crowds with slightly better weather.
Overall, weeks around this general vicinity are usually good as well as long as you're avoiding peak seasons, as those are very hectic for both the first-time visitor as well as the veteran Disney goer.