I feel like the RS-1i sound similar (I'd use the word "radiant" but "aggressive without being bright or forced" also works for me) via my Musical Fidelity; which is not tubed. I felt like the CIA amp did a good job with the SR-225, when I had that setup. In general I'm not a big proponent of "amps are a night and day sonic difference" though - so yeah.
Regarding Beats Pro versus RS-1i; where to begin.
The Beats Pro are closed, and fairly closed-in sounding as a result; they aren't as comfortable (clampier + heavier), but the build quality doesn't feel that bad. Sound is different - both are able to be punchy, but the Grado are more refined and airy, perhaps a bit faster, less bass slam, better midrange voicing, and (imho) more engaging and textured bass. The Beats Pro are more of a "wall of sound" - they aren't really distorted though, and play LOUD with ease. I think that the RS-1 are a "superior" headphone, but it's a tough comparison - you're talking open v closed, different sound sigs, etc (before you even get into the "Beats suck" discussion). The Beats Pro are probably a better choice if you need something to truck around on the bus, or if you want to play them stupendously loud (which will wreck your hearing, so I don't suggest it); the RS-1 are very likely to give you a better sonic experience assuming you're going to use them at home or somewhere else that's relatively quiet and where they don't have to stand-up to a lot of rough handling. Cost is also a factor (the RS-1 are nearly double the money), so that should be considered.
Overall I think you won't find a much better open headphone than the RS-1, especially around their price, but I think you can find a better closed headphone than the Beats Pro, especially around their price. If that helps.