I wasn't sure how to explain my decision to move from Evangelicalism/Anglo-Catholicism (when people asked my denomination, I'd say either "miscellaneous" or, if they were familiar with Facebook, "It's Complicated"), but perhaps poaching (can you "steal" questions?) from the National Catholic Reporter's interview with recent-convert-from-Evangelicalism Francis Beckwith would be a good place to start.
You spent quite a few years in the Evangelical and Anglican worlds. What could Catholics learn from Evangelicals and Anglicans?
Whew, let me go find my soapbox. In my view, a lot. However, let me back the truck up for a moment.
I believe that it is counter-productive for Roman Catholics to consider Evangelicals and those in Mainline denominations "Protestants." The term conjures up visions of laborers striking for better pay and working conditions who will return to the job once their conditions are met, or at least their grievances aired. This simply does not reflect the reality of the situation. The workers have left and started their own factory using the knowledge they gained from their old jobs. In my view, a much better analogy would be the Kingdom of Judah and the Kingdom of Israel in the Old Testament. "Protestants" are now more-or-less self-sufficient (whether they're mortally wounded or not and don't realize it is another question entirely), and certainly independent. I would even go so far as to question whether these "tribes" have been ripped away from the Throne of Peter as a result of the wickedness it allowed and even condoned. The Lutherans and Anabaptists had some pretty legitimate gripes, with the Roman Catholic Church seriously abusing its position and definitely failing to properly educate the people in their faith. The Anglicans had much less of a claim, but Henry VIII had a reasonable expectation that the annulment he sought would be granted, as this seems to have been the common practice at the time and if it weren't for the intervention of his wife Catherine of Aragon's nephew, the Holy Roman Emperor Charles V, it probably would have happened. As far as I can tell, all "Protestants" ultimately trace their origins to these three groups. The Roman Catholic Church (which, as a reminder, I'm aligning myself with) bears a huge responsibility for the many, many souls it has let slip away due to corruption, ignorance, apathy, and pride. Back to my point, though, these groups aren't protesting anything. It's not even an issue, and if anything, they believe the Roman Catholics should re-unite with them and not vice-versa. Most of their believers come from a long line of non-Catholics and they no more "reject" or "protest" Roman Catholicism than I reject or protest British citizenship. Rome has already had its Counter-Reformation, and even signed a joint statement with Lutherans saying that they believed the same things on Justification. Yet, the Lutherans aren't seeking reunion, and this should tell Roman Catholics something. Calling such folk "Protestants" simply distracts from the reality of things.
Before saying what can be learned, let me say that nothing essential should ever be compromised. Doing so is a plan for suicide. At the same time, it's important to recognize that simply being vested with doctrinal and institutional authority does no necessarily mean that other Christian groups aren't doing things better. Also, I'm only listing the positives for the two groups; there are plenty of "make sure you don't fall into this trap" things that could very easily be mentioned.
The biggest strength of Evangelicalism is its emphasis on knowing the Bible. While I believe this comes from the mistaken idea that the Church is a product of the Bible and not the other way around, this study is a powerful tool. Roman Catholics have a reputation, which in my experience is often well-deserved, of not knowing what the Bible says. While Evangelicals can often discuss how the prophecies of Isaiah relate to Christ, Roman Catholics are confused by the term "Pauline" to describe half the Epistles. Evangelicals are known for having weekly Bible studies, for having (often overly-long) sermons that delve deep into the original meanings of words in Greek and Hebrew, and for being able to quote verses of Scripture (even if selectively). This literacy of Scripture is a powerful tool, as it is described as a sword and a shield by the Bible in Ephesians. While the Roman Catholic Church has a strong tradition and arsenal of prayers, they tend to be more subtle and defensive in nature, if that makes sense. Scripture is more offensive (in both ways) than prayer, and sometimes it's the right tool for the job.
Other strengths include more of a willingness to utilize updated methods to express beliefs; "praise and worship" music is probably the best example of this, along with the related genre of Contemporary Christian Music. To be fair, much of this is dreck. Again, I don't know the source, but it's been pointed out that 90% of just about anything is junk. Not all Classical composers were Bachs, Beethovens, Mozarts, or Palestrinas (yes, I'm merging Renaissance, Baroque, Classical, and Romantic; sue me); these are the ones whose work has endured for centuries. There are some jewels to be found among these, though. In particular, I would recommend Casting Crowns and Rich Mullins for P&W and Five Iron Frenzy, Audio Adrenaline, and Relient K for CCM (these are all kind of 20-something guy -type groups, though). These influences, along with that of a lot of Evangelical writing, seem to be making inroads among Roman Catholics.
As for Anglicans, their biggest strengths are in what is sometimes seen as a more "tasteful" and "reasonable" form of Roman Catholicism. Although this is often mocked as being style over substance, there's something to it. Consider aesthetics. Roman Catholic styles look to outsiders as gaudy, tacky, dated, and somewhat feminine. Episcopalian styles are more tasteful, well-appointed, classical, and neutral. I know that's a subjective call, but think of plaster statues, glow-in-the-dark rosaries, and pastel vestments, and then consider stone buildings, the Book of Common Prayer, and bolder colors. Some of this is culture-based, too: most English-speaking "Protestants" are of Anglo-Saxon stock, or at least mindset. To a large extent, they look at Roman Catholicism and say "I don't see my culture reflected there." It's seen as a faith for Irish, Italian, and Mexican people. To my mind, this is why some sort of provision for a distinct body of Roman Catholics with an Anglican flavor would be worth having. In English-speaking countries there is a huge, huge cultural barrier to be overcome for Evangelicals and Anglicans to pursue Roman Catholicism. Seriously, the view is almost that of people swinging at a pinata full of Communion wafers. As battered as the reputation is, Episcopalians still have a kind of street cred as being more-or-less reasonable. Such a group would provide a kind of bridge, as it would be doctrinally sound but without the "ethnic" baggage (yes, Anglo-Saxon is an ethnicity, but we're speaking subjectively here). A spoonful of sugar makes the medicine go down.