I tend to follow the speculation that "warp drive" is the thing that makes the ship go FTL, the "warp core" is the thing that creates "warp power" used by the "warp drive" and "Matter/Antimatter" is the fuel "burned" to create "warp power" in Federation starships. And the reason the Federation uses M/AM is that when matter and antimatter annihilate, 100% of the fuel's mass is converted to energy, which is then available to be turned into "warp power".
Furthermore, I tend to follow the speculation that "impulse" is the newtonian rocket used to travel STL and "impulse power" is where the the exhaust gasses get their high exit velocity. In the Federation this comes from fusion --at least in part.
It seems perfectly plausible that one could turn on a "warp drive" using "impulse power". With the above speculations, Kirk could have powered his warp drive in "Where No Man Gone Before" with fusion power to travel to Delta Vega from the galaxy's edge. And of course, his difficulty with this would be that the amount of mass released as energy in even the best fusion-fuel-cycle is less than 1%. And most of the speculations I've read say the Federation uses a deuterium-deuterium fuel cycle that translates only 0.0973% of the fuel-mass to energy.
That would be an enormous downgrade in available power for the Enterprise, easily making it impossible for her make it home in WNMHGB without dilithium.....I mean, a power source that converts a thousand times less fuel-mass into energy is going to need a thousand times more fuel-mass to power the same warp drive to the same warp factor; conversely, you'll get one thousandth the power if you pump the same fuel-mass through. Indeed, if you take the TNG Tech Manually seriously, than it takes approximately a thousand times more power to cruise at 6.0wf as it does to cruise at 2.7wf. So if you have the same fuel-mass flow rate while using a deuterium-fusion fuel cycle that gets one thousandth the mass-energy conversion as M/AM, than you'll travel at 27c rather than 320c. (TNG warp scale: apples to apples.) That's the difference between making it home and not --precisely Kirks problem in WNMHGB.
Using these same speculations --and I have no delusions they are otherwise-- and further assuming Romulan BOPs are fusion powered (though not necessarily by the above mentioned deuterium fuel cycle), the BOP in "Balance of Terror" would have had terrible constraints on her power use. Her cloaking device, warp drive and plasma weapon would all use up power --and burn through fuel-- very quickly. Add to this the extra space the weapon would take up and perhaps her fuel supply was even further constrained. Thus she would be terribly hobbled compared to a standard BOP, such as the one from "The Deadly Years". Thus a standard BOP could closely match the Enterprise in speed when the BOP of BoT could not dream of it.
Of course, if the cloaking device had worked as well as her crew expected, then the design compromises found in the BoT BOP would probably have been worth it. But it didn't, so it wasn't.
(Na na na na na na, BoT BOP!)
But that begs the question: If fusion is so inferior to M/AM as a power source, why would the Romulans use it? I think the answer comes down to infrastructure. Antimatter needs to be made. Fusion fuel can be collected. A military that relies on stealth to win doesn't want fuel-supply-trains giving away the position of vital outposts, or forcing home a long-term infiltration vessel when they could be monitoring the enemy. Fusion would allow a vessel to be permanently energy self sufficient, especially if its fuel cycle took regular hydrogen and fused it into helium or heavier elements.
(The best fuel cycle I've figured out is fusing 54 protons over 18 reaction stages into Iron54, a mass to energy conversion of 0.935%. However, eighteen stages seems a bit much and the reactions were cherry-picked for energy output, not likelihood or ease of fusion. A more likely scenario, relatively speaking, is 56 protons fused over 7 stages to Nickle56, releasing 0.879%, still a considerable amount for fusion. But for now I'll stick to fusing hydrogen to helium in 3 stages, releasing 0.712%.)
Hydrogen is everywhere and can be collected from any gas giant's atmosphere, any water world's oceans or even any rocky world's lithosphere. But far more exciting is combining the bussard ram scoop with a hydrogen fusion warp core. If you assume the interstellar medium has an average density of about 1 atom per cubic meter and that 89% of that is hydrogen --currently, not a bad estimate-- then to keep the Ent-D at warp 5 cruise, you'd need a collection field that swept up an area approximately the same as a circle with a 20km diameter. That's pretty big, but not huge. And it's certainly do-able by Trek standards. A Rom TOS BOP would probably need much less than the Ent-D...How much less? Naturally, I don't know.
But what that basically means is that a BOP would only need enough fuel to initiate 1wf, then she could turn on her ram scoop to maximum and accelerate. The interstellar medium of any particular volume would only be able to support a particular warp factor for a given scoop area; and any power plant would have a limit on the fuel mass it could intake. But the vessel's range and acceleration would not be limited by the amount of fuel carried, unlike a M/AM powered warp core.
Problem: That was directly contradicted in BoT when the Romulan commander spoke with concern that their fuel was low. ... Except for two things.
First, the BoT BOP had been going in and out of warp, firing their weapon, engaging and disengaging their cloak, and generally sucking up power like the dickens. Well, all the way to warp 7, getting over an integer warp threshold takes at least an order of magnitude more instantaneous power than cruising at that integer warp factor. So his concern might have been that they were getting so low on fuel they would have to stay at a lower warp factors while they scooped up enough fuel to jump over the next warp threshold, and thus wind up staying in the neutral zone too long....This seems a bit of a stretch.
Far more likely, IMHO, is that a field capable of collecting the needed gas would be a powerful beacon that would gave away their position at long distances, something they couldn't afford until they were in their own territory.
By the way, once you have an artificial quantum singularity and you can convert 100% of the mass you extract from the interstellar medium, your scoop diameter need not exceed 1km to support your warp flight until the warp 9 regime. (Again, assuming you use the same amount of power per cochrane as Ent-D) Once again, pretty awesome on the no-infrastructure-needed front.