Now that I understand the effect of expansion on tech, that makes great sense. Thanks.
From some of the discussion, I wonder if there is a difference between conquered provinces and colonial provinces in the 1st 50 years (until you gain a core) and thereafter regarding economic/tech development? Something tells me that rapid expansion by conquest is easier than rapid expansion by colonizing. If so, why?
The answer is that rapid conquest can potentially yield you some nice wealthy high income provinces early. Colonial provinces usually start out poor and then come into their own later in the game, it's a longer term investment.
As was mentioned in the discussion about technology, the cost of technologies scale upwards based on the number of provinces you have. The result is that, as a rule of thumb, instead of worrying about how many provinces you have, it's more useful to think about your average income per province. If your conquest or other expansion is increasing your average income per province, you will be more productive overall and should be researching faster, all other things being equal, whereas if your expansion drops your average income per province, you will be researching slower. Incidently, this is why some of the OPM's do so well in the tech race, especially the good traders like Lubeck, their income per province is usually very high. This also means that conquering that crappy base tax 1 province with fish for their trade goods is usually not worth it, at least from a purely economic perspective (note: vassalize it instead for extra income with none of the extra costs, vassals are great!).
So, when trying to decide whether to go on an early conquering spree, you want to consider the income generated in the provinces and go for the high-income targets. Next, consider the negative modifiers to income that might occur if you conquer it:
-non-core?
-different religion?
-differnt culture group / non-accepted culture?
These are the most important factors, if the answers are yes to these questions the province is probably going to be a drain on your economy and you should think twice about whether its worth conquering, or you should consider making it a vassal instead. On the other hand, if it's a high income, same religion, same culture group (or accepted culture) province, it's definitely a worthwhile target. If you can somehow get a core on the province through event, mission or spying, even better. This is actually where the culture map and religion map is very useful, expanding to the border of your culture group or religion is often a very good idea, and only after that go colonizing.
The same principles apply to colonizing by the way, you want the provinces with the lucrative trade goods like sugar, spice, etc. That colony on some island rock full of fishermen, however, is probably not worth it, unless for purely strategic reasons, e.g. as a naval base to launch expeditions to more lucrative areas.
Returning to your original question, it seems odd to me that you are having trouble keeping up, especially with Spain at 34% inflation (I did read that right, didn't I, they are at 34% inflation and you are around 3%?). Is your high income new, i.e. did they have higher income than you for most of the game? If so, you should start catching up over the coming decades. Other factors to consider is just that they have much more lucrative provinces than you (which sounds likely), are religiously unified, are culturally unified, have a higher proportion of core provinces, have less revolt risk, have higher stability, have more/better buildings (market, refineries, etc), have lower inflation, have better sliders, and better economic national ideas than you (I think this is a pretty exhaustive list of all the factors that affect income, but I may have missed one or two!). If none of these explain it, then I'd suggest it's probably just a 'need more time to catch up issue,' so keep pushing your average income per province upward and you should start catching up and then moving ahead.
Usually in my expansionist games I only really start moving ahead in the tech race after my income is way way higher than my closest competitors. Just being at about the same income level as them is often only enough to stop yourself from falling further behind, maybe to also start slowly closing the gap. It's certainly enough to stay caught up in a couple out of the six tech areas, so be sure to prioritize which tech areas are most important to you and focus your research dollars there.