Your description of the model functioning is correct. Structurally, both models are representatives of the so called kernel methods. As such they are very similar, the same in many cases.
What is completely different between the two methods is the way the kernel centers and linear coefficients are derived. My experience with SVM is limited, so I cannot go into the details of the training method, but in summary is consists of solving the constrained optimization problem of finding the best approximation subject to maximum coefficients. This is achieved by linear programming or through a method like Sequential Minimal Optimization (SMO).
Training RBFNs, on the other hand, is a different story. Identifying the kernels is done in one of two ways usually; either by using a clustering algorithm or by using Orthogonal Least Squares (OLS). In either case, the linear coefficients are identified as a second step through least squares (LS).
The differences in training mean that, even through in structure the resulting models may be the same, in functioning they will most probably be completely different, due to the different fitting procedures.
J. C. Platt, “Fast Training of Support Vector Machines Using Sequential Minimal Optimization,” Adv. kernel methods, pp. 185–208, 1998.
J. Moody and C. J. Darken, “Fast Learning in Networks of Locally-Tuned Processing Units,” Neural Comput., vol. 1, no. 2, pp. 281–294, Jun. 1989.
S. Chen, C. F. N. Cowan, and P. M. Grant, “Orthogonal least squares learning algorithm for radial basis function networks,” IEEE Trans. Neural Networks, vol. 2, no. 2, pp. 302–309, Mar. 1991.