In terms of overall gaming performance, the graphical capabilities of the AMD Radeon 8500 Series 64MB are significantly better than the Nvidia GeForce 4 MX 440.
The 4 MX was released over a year more recently than the 8500 Series, and so the 4 MX is likely to have better driver support, meaning it will be more optimized for running the latest games when compared to the 8500 Series.
Both GPUs exhibit very poor performance, so rather than upgrading from one to the other you should consider looking at more powerful GPUs. Neither of these will be able to run the latest games in any playable way.
The 4 MX has 64 MB more video memory than the 8500 Series, so is likely to be slightly better at displaying game textures at higher resolutions. However, overall, the 8500 Series has superior memory performance.
The 8500 Series has 5.6 GB/sec greater memory bandwidth than the 4 MX, which means that the memory performance of the 8500 Series is marginally better than the 4 MX.
The Radeon 8500 Series 64MB has 4 Shader Processing Units and the GeForce 4 MX 440 has 2. The two GPUs are based on different architectures, but deliver an equivalent shader performance. To compare, we must continue to look at the memory bandwidth, Texture and Pixel Rates. In this case, we sadly do not have enough data in this area to complete the comparison.
The Radeon R200 is the second generation of Radeon graphics chips from ATI Technologies. The architecture features 3D acceleration based upon Microsoft Direct3D 8.0 and OpenGL 1.3, a major improvement in features and performance compared to the preceding Radeon R100 design. The GPU also includes 2D GUI acceleration, video acceleration, and multiple display outputs. 'R200' refers to the development codename of the initially released GPU of the generation. It is the basis for a variety of other succeeding products.
GeForce4 MX 440 is part of the GeForce4 GPUS released by NVIDIA in 2002. Only compatible with DirectX 7 or less and with a max memory of 128 MB, it can't play today's games.
Still, games before 2003 should be fully playable at max settings....