The lack of a union with the Habsburgs does bring important changes to Portugal, the most important being that Portugal will not be entangled in the conflicts with other European countries. For the past months, I have been going through the archives of Portugal's diplomatic correspondence housed in the National Archives of the Torre do Tombo. During the 16th century, Portugal was careful to avoid conflict with any European country as this was seen as damaging to the country's main source of wealth, trade. The Portuguese monarchs even wrote letters to various rulers, particularly in France and England lamenting the state of war and calling for a unity of Christian princes to take on the Saracens. Another theme was Portugal's attempt to protect a trading monopoly in the areas which in considered hers by virtue of the Treaty of Tordesillas. Prior to 1580, the French were the primary violators, attempting to establish settlements in Brazil and French privateers often plundering ships bringing goods from the Low Countries to Portugal and vice-versa. What is interesting to note is that the French ambassadors correspondences were generally very apologetic and blamed corsairs, which the crown had no control over. With regards to the English, Henry VIII sought to have English sailors accompany the Portuguese to the Indies, but this was rebuffed. The appearance of English traders in Guinea led to a protest by the Portuguese Ambassador, and eventually the English agreed in 1576 to not trade in these lands.
With the arrival of Philip II on the throne, Portugal was now at war with England, France and the United Provinces at various times, leading to an unprecedented number of attacks on Portugal and its empire. Among the most significant were those from the United Provinces, particularly from Holland and Zeeland. Prior to the Dutch revolt, Portugal had become economically dependent on Flemish capital and shipping, particularly from Antwerp. Flemish and other Northern European banking houses lent large sums to the Portuguese Crown and to Portuguese merchants. They also provided much needed grain to Lisbon, as the Portuguese hinterland was unable to feed such a large city. The repression by the Duke of Alba would lead to a flight of capital and merchants from Antwerp, to other cities, particularly Amsterdam.
Also, until 1595 Dutch ships came to Portuguese ports, bringing with them much needed grain and industrial goods, and acquiring sugar from Brazil, spices and other goods from the Indies to resell in Northern Europe. Importantly, in 1578 and 1579 some 130 ships per year also arrived to acquire salt from Setúbal, it was re-exported throughout the Baltic region to preserve herring. When the Spanish placed an embargo on Dutch trade with Spain and Portugal in 1595, the Dutch sent their first exploratory missions to Asia and the Americas.
If Portugal remains in the hands of the house of Aviz or an independent dynasty I imagine things might go a bit as follows. After the 1520s, the costs of war in North Africa (Morocco) were eating much of the profits from the Asia trade. By the 1550s, the Portuguese Crown abandoned many of its smaller ports and spent large fortunes fortifying its remaining holdings in Morocco. Nevertheless, these were a constant drain on the treasury, but they could not be abandoned as Ceuta and Tangier were seats of Bishoprics. Also, after the 1550s, Portuguese trade with Asia is growing and will probably continue to grow beyond the 1620s (when it decreased IOTL), but increasing competition from the Levantine Trade via the Red Sea will continue to lead to more private merchants trading, rather than the crown's official Carreiras da India.
In 1531, the Portuguese crown began allowing foreigners to trade with Brazil in return for a 10% duty paid in Lisbon, and this had led to Dutch traders being active by the 1570s. By the 1580s, private merchants of Portuguese merchants houses, whom often were raising funds in Northern Europe were dominating the trade with the Indies as well. Without the Dutch directly attacking the Portuguese, I imagine more Portuguese forts/strongholds in Asia will remain in Portuguese hands. On the other hand, the increasing trend of inter-Asian trade dominating the Estado da India, with local merchants reaping most of the rewards, rather than Lisbon is a trend that will probably continue. I imagine that Dutch and English traders might be able to eventually buy concessions to trade with Portuguese ports in Asia, relieving the foreign powers of having to spend large sums on defence.
One of the major problems with Portugal was that the largest families tended to invest their profits from trade in entailed estates and land holdings rather than in new businesses, so I still see a gradual decline occurring. As for foreign competition, the VOC might not be formed at all. It is important to note that prior to 1602, individual Dutch traders went to Asia and were able to obtain larger profits than in the post-VOC period. Asia was so large and the Portuguese were spread thin, so they simply avoided the Portuguese ports. Some areas like Sumatra, Bengal, the Coromandel had minimal Portuguese presence, and other powers could easily establish a presence there. It is often forgotten that the VOC had very poor returns for its investors until the late 1630s, and the original capital of 800,000 guilders had declined to around 560,000 in 1621. The figures on the outflow of capital from the VOC are shown. However, it is important this this was seen as a necessity, because of the state of war with the Spanish Crown. Without such a war against Portugal, the States-General would probably not want to antagonise a friendly trading partner with whom they enjoyed a financial beneficial relationship. Additionally, Flemish manufactured goods, such as clocks were already being traded in Asia by the Portuguese by the 1570s. The Dutch might simply see it fit to have the Portuguese assume the cost of arming ships and engaging in bribes and warfare with local Asian potentates, while reaping the benefits. Even the diamond cutting, with the Portuguese bringing diamonds from the East, largely bypassed Lisbon where the uncut stones were shipped to Antwerp making it the leading it to replace Venice as the leading hub of the gem trade in Europe by 1570.
Though a decline relative to its peers was probably still inevitable, I can see some major benefits to the lack of the union. Without it, Portugal will undoubtedly spend less on defence, allowing it to limp along as a middle power for a while longer. Its decline might be much slower as the navy probably remains among the top three in Europe at least until 1700. Also, as a result of the Restoration War against Spain between 1640-1668, Portugal had to spend huge amounts of its budget on recruiting armies, including paying mercenaries In desperate need of a ally, Portugal also spent a vast sum marrying off Catherine of Braganza to Charles II of England, with a dowry of 2 million cruzados, along with the cessation of both Bombay and Tangier. Additionally, peace with the Dutch was only reached in 1661 (only ratified in 1663) where to end the war, Portugal had to agree to indemnify the United Provinces with 63 tonnes of gold over a sixteen year period. This was to compensate for the reconquest of Brazil, Angola and São Tomé. Meanwhile, the Portuguese were given nothing for the loss of El Mina and Axim, Ceylon, Malacca, Cochin, the Moluccas and Macassar, not to mention the blockades and attacks on Lisbon, Goa and other ports. Desperate for peace and recognition as a legitimate ruling house, King João IV and, after, his death his wife Luisa acquiesced to these demands.
Another important effect on world history is the sugar trade. By 1580, Brazil was supplying the vast majority of Europe's sugar. Also, by 1580, the first sugar refineries were operating in Amsterdam, where the Dutch would reexport this product to other Northern European ports, including in England. In 1621, the Spanish Crown reimposed its embargo on trade with the Netherlands, leading to the formation of the WIC. It would eventually occupy Northeastern Brazil to acquire sugar. Without an embargo, there probably is no such attempt, but importantly this might delay the transition from tobacco to sugar in the West Indies. The retreat of the Dutch from Brazil and the exodus of Portuguese Jews whom had come to Brazil in the 1643 led to some settling in the French and British West Indies, providing the knowledge to establish a sugar economy in those islands. I imagine the English and French in particular would still acquire these islands and grow sugar, it might be delayed by a decade or two.