Expanding the question, this effect wasn't restricted to the early modern era, if I'm not mistaken. The British Empire had a huge deficit with China the 19th century, and the Roman Empire also had one with ancient India. Why is that?
The simple truth is that Europeans had very little to offer.
Recent work by global historians has really highlighted how the various squabbling European dynasties from the Middle Ages to about the 17th century were a total backwater in terms of global trade. These European dynasties not only lagged far behind their counterparts in (modern-day) India and China (in particular) in areas like technology, education, and political structures & centralization, but also (and this is the key) in economic sophistication.
Remember that the Middle Ages, European economies were fairly close to subsistence for the vast majority of people, and the surplus goods that were produced were fairly 'rough'--meat, timber, cloth, cheese, wine--which were bulky, difficult to transport, and often just as cheaply-produced in India or China. Sure, blips like the Hanseatic League or the Champaign Fairs looked impressive to Western historians for awhile...because they looked impressive to the rather yokel-ish Europeans who gawked at the 'wealth' seen there. But the Champaign Fairs were (quite literally) pig-fests compared to the sophisticated markets in the near and far East that dealt with far more refined and specialized goods.
In fact, the Indian ocean and "silk road" trade networks had not only existed for centuries--which fostered the niche-producers of luxury goods spices, silks, and intricately-carved ivory work--but (and this is especially true of the Indian Ocean routes) they also moved large quantities of basics goods like grain and timber.
Here's a case to illustrate what I mean: when Vasco da Gama finally made it around the Horn of Africa in 1498 and (after picking up a very experienced Muslim pilot in East Africa--a professional pilot who was part of the trade network) made it to the Malabar Coast of India, he managed to get an audience with the King of Calicut—the Samoothiri—who at first gave da Gama a lavish welcome:
and, as was traditional, Da Gama presented the Samoothiri with gifts… consisting of a few hats, a few rough-sewn cloaks, and a chest of glass beads. Glass beads! The Samoothiri mocked him outright—why no gold? no silver? His advisors whispered: perhaps this is no real ambassador—merely a pirate. How right they would prove to be.
The truth was, the Portuguese--or the Europeans overall--had little that was desirable to the East. None of the European "surplus"--animal hides, beer, that sort of thing--had even the slightest bit of appeal in the Indian principalities or in China. Cheap goods, practically one step above raw material, certainly not worth importing over 1000 miles.
The Europeans were also quite far behind in most areas of technology as well.
They had a few breakthroughs: they were very good at working with metal, to make church bells (EDIT--out of bronze and brass) and breastplates out of steel and the like. And this meant that, when gunpowder came into vogue, only the Europeans had the skill to cast cannons flawless enough to not explode and kill the gunners. And they also had a great deal of skill with ocean-going vessels--because hauling a bunch of fish out of the North Sea (when your land is too poor to support your people) demands some serious shipbuilding. Finally, they'd been on the 'downhill' side of a thousand years of plagues, and lived in appallingly un-hygenic conditions (compared to Asians), so they were resistant to the diseases they carried with them.
(note: this is a VERY different argument from Diamonds' Guns Germs and Steel, because Diamond wants to 'naturalize' these European developments as 'advantages'--when, in fact, in the 16th century, things like guns, disease-resistance, and steel were culturally created artifacts that came out of (relative) poverty and misery (like, a 1000 years of devastating warfare, anyone?)
19th and 20th century historians tended (like Diamond) to project European superiority backwards in time... which greatly distorts the historical record.
What the Europeans were able to do, then, was accidentally leverage their inherent disease resistance in the new world into a sort of bio-warfare, which devastated indigenous communities, and 'opened up' vast areas of productive land for exploitation. The Spanish--finally!--came across something of interest to the Indians and Chinese, namely gold and silver...which they extracted using slave labor--and the enforcement of slave labor was enabled by their superior military technology.
In the Indian Ocean, meanwhile, the Portuguese used their only advantage--sea-born military technology (cannons on caravels) to set up a protection racket... which made them wealthy, but eventually fragmented the Indian Ocean networks.
Europeans would eventually leverage their 'science' (itself a rather odd outgrowth of European religious obsessiveness--monasticism--but I digress, and that's a really different argument) into machines... and as Michael Adas argues so convincingly in Machines as the Measure of Men...it is only when Europeans had superior machine-technology in the late 18th and early 19th century (note: they were still massively inferior at producing refined goods like silk or fine textiles) that Europeans began to insist on 'measuring' the worth of a society by its ability to produce those self-same machines...not to create sophisticated porcelain or weave beautifully-subtle silks.
So, when the British still had a massive trade deficit with China in the early 19th century, due to their addictions (tea, mostly, but for many other refined goods, unable to be produced in the clumsy water-powered or steam-powered factories), they first paid for in kind with their own drug (opium), and when the Chinese got a bit tetchy, the Europeans were forced to resort to good old military superiority. (the Nemesis, a steam powered warship, shooting up the trading junks on Chinese waterways in the 1840s, for instance.)
In the Very Big Picture, then, the Europeans, in short, were a lot like the mafia: they came out of poverty, preyed on much more sophisticated and elaborate economic communities by being thuggish, thereby levering themsevels into wealth, and then using that capital to diversify into legitimate business enterprises.
A great overview of this is Robert Marks, The Origins of the Modern World, who has built on sophisticated work by William McNeill, Ken Pommeranz, and many others.
EDIT: some are saying I'm being a bit harsh on the poor Europeans... and maybe I'm 'overcorrecting' a bit too much.
So here's one "European" invention that was not war-related and that was nonetheless unmatched: the university.
The first universities (Bologna, Paris, Oxford) came out of the Middle Ages, where they were glorified church-law schools--though church law was admittedly a big deal when the church owned about 20% of everything.
But by the 15th century, they had taken off on their own intellectual trajectories... and by the 18th century, they were incredibly impressive conglomerations of Very Smart People working to advance knowledge.
By the 19th century, the European/Western University was an unparalleled site of research and learning. There was literally nothing like it in any other part of the world. That's why, by the way, that the writing of history has been so western-biased for so long: Europeans were the first and (by the 19th century) unmatched historians of the world, serious professionals when everyone everywhere else was still amateur-hour.
There are other examples of European innovation too, of course. But it's good to keep in mind the sober realities of European weakness as well as strength, to get an appreciation for the subtleties--and sheer ironies--of history.
EDIT2: also, clocks. watches. any sort of time-piece. Europeans loved their accurate time measurements. Goes back to--you guessed it--monasteries, where monks invented water-clocks so that prayers could be attended in a timely fashion. From there, it was one step up to the actual clock-towers... then, the little metal figurines that chase each other arounds... then, ultimately, pocket-watches. All of those teeny-tiny gears were just stunning... And the Chinese ate them up. Also, did I mention telescopes? Hats off to Galileo for that one...
EDIT3: also, I know that I more or less skipped from more-or-less Medieval Europe to the early 19th century... but that's because the questioner framed the question as an Age of Discovery (16th c.)
Here's my take on the Consumer Revolution of the 17th and 18th c., from my answer to one of the questions below:
EDIT4: a number of people are really perplexed by the cannons-from-church-bells argument, so I went digging for it: my source for that is McNeill's classic "Age of Gunpowder Empires" essay, found in the edited volume by Adas Islamic and European Expansion (p. 106), but also widely available online as a pdf. McNeill talks about how the experience with casting bells as a single block of metal (using molds) with bronze and brass bells was easily adapted to casting single-mold cannons with iron.
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Is the premise of this question correct? I was under the impression that the Age of Exploration was driven more by the slaves than spices, something that tends to be glossed over in high school history.