MATLAB: Fourier transform vs $\tt lsim$, results don't match
To perform the FFT approach correctly, you will need the FFT length to be at least $N_U + N_H - 1$ where $N_U$ is the length of $u$ and $N_H$ is the length of $h$.  

It looks like $h$ is an IIR filter, so the FFT approach probably won't work well.

The Mathworks has a nice example comparing circular with linear convolution.

IIR filters are best implemented in the time domain. For their specs, they are usually much lower order than FIR filters and so we can afford to implement them directly.