The Federation theoretically should be huge and powerful. They are spread across 8000 Lightyears, have 150 member worlds and very likely thousands of colony worlds.
Huh uh. The line in First Contact was 150 planets. Period. No qualifiers.
Except it obviously referred to full members. For an example, I watched "Schizoid Man" today. The world referred to as "Graves' Planet" is pretty aptly-named, because he is literally the only guy on it (technically, Karina Briannon [however it's spelled] is also an inhabitant, but she is no guy, a fact upon which the story turns). Do you really think Picard was counting a crap world like that just because it was in Federation territory and occuped by however few Federation citizens? Indeed, if you went and listed every planet ever mentioned as a Fed-controlled world or colony, I suspect it would actually exceed 150.
If you want to take the man literally, there would actually be at least 150 planets in a mere 15-30 star systems. Sol accounts for eight by itself. And we've certainly seen more than 15-30 Fed systems (heck, we've seen more than 15-30 Fed sectors!). Should we count systems that have no life at all as well? They still contain "planets."
Interestingly, if you do interpret the dialogue to be completely literally, you can't actually count Andor, a Federation founder.
That said...
I Am Legend said:
For any other civilisation with equal technology to counter the Federation and be as strong they would need the equivalent number of worlds and territory.
Why? Since we know the primary power source of all major empires is antimatter (except apparently Romulus, which relies directly on black holes) we also know that terrestrial resources are pretty much worthless, barring dilithium, which does not at all require a native population to exploit. In fact, obliterating any native population on a dilithium-rich planet would probably make it easier to exploit. A lot of stuff is apparently run by fusion, I'll grant, but fusion reactant harvesting doesn't exactly require "fully developed planets," either. It just requires access to water, a gas giant, or given the capabilities of Star Trek science, perhaps even the corona of a star. Again, dead natives would make this resource easier to exploit. As for metals, that could be best exploited without ever venturing down into a planetary gravity well. Plenty of metals in asteroids. Klingons could be raping our asteroid resources as we speak and we wouldn't necessarily even know it.
Conversion of raw materials into useful products, of course, is almost entirely automatic, due to the prevalence of replicators. Even in the 23d century, a combination of proto-replicators with simple automated factories were the likely means of production.
Edit: I would like to point out at this juncture that I fear someone might bring up the notion that any given planet's agricultural resources are worth dick, even locally. This is a completely spurious idea. Replicators are everywhere, and you don't need grain to feed into it. A comet will do just fine for bulk carbon, hydrogen, and oxygen. Nitrogen is not exactly hard to find, and the other elements biological beings need are not ordinarily rare, either. So crude oil, ashes, or the methane lakes of Titan would be excellent sources of food. One's own crap, shed skin cells, and dead bodies will suffice, really, as long as you can add sufficient energy to it to make it palatable again, and don't need to increase one's biomass, i.e., make more people.
All this means that the raw industrial potential of an interstellar government is by no means reliant on a large population.
What a large population provides, and this is borne out more-or-less by the Federation ordinarily being significantly more advanced technologically than its competitors, is the intellectual resources of greater numbers. If the UFP has a population of a trillion, and the Klingon Empire an effective population of 50 billion, each may have access to the same number of energy sources and mineral resources due to the vicissitudes of history, but one of them is going to have access to a proportionally greater number of scientists, engineers, strategists, and administrators.
The inevitable pressure, therefore, is to bring the client peoples closer into the mainstream of the society and the economy. This is what the Dominion apparently did, to great effect. The Klingons, Cardassians, and Romulans, by contrast, did not, and therefore wound up, respectively, a junior partner, a conquered enemy, and the Trek equivalent of a failed state dealing with a faintly retarded class war.
(Interestingly, and counter-intuitively, there is every indication the pseudo-libertarian Ferengi would welcome cooperation with other species for mutual technological benefit, and hence are probably at the end of the 24th century the second most powerful government in the Alpha Auadrant.)