Converting gold to research means that gold can be used for just about everything in the game. Gold can buy units, upgrade units, buy buildings (=happiness), buy tiles, buy culture and buy food (CSs). With the option of RAs gold can buy techs as well. This flexibility means that building gold in a city is almost always the best option, except for a few production/GP wonderhouses. Since you're building gold anyways, the choice is whether to spend 250g on a new tech or 250g to, for example, upgrade a few units...this decision, of course, depends on your situation.
Or you could spend 250g on upgrades, capture and pillage a couple cities, and use the pillage gains on an RA...
I'm not sure why you would decline an RA even with a more advanced AI. You're both getting a free tech. Unless the game somehow skews techs in the AIs favor (I have no idea how that would even be possible?) at worst the RA makes you both equally more advanced.
What I can't figure out is if the RA gives you both the same tech? That would seem to make the most sense...but I don't see it spelled out anywhere. I suppose if the RA is giving you Gunpowder and your opponent Flight, then no, that probably isn't a good trade.