AVI has been the long standing format to save and deliver movies and other video files. It is a multimedia container format which contain both video and audio data in a file container for sync play audio and video.

ASF is Microsoft's proprietary digital audio/digital video container format, especially meant for streaming media. ASF is part of the Windows Media framework. just specifies the structure of the video/audio stream.

1: AVI does have some of the advanced capabilities that are available in ASF.
2: AVI is thought of as a container format with the ASF is a container format for streaming media.
3: AVI is a more universal format (Linux can play it) but requires you hunt down software that will play it

Other Comparison: MOV vs WMV / 3GP vs 3G2 / MP4 vs FLV / MKV vs MP4 / MP4 vs MP3 / 3GP vs MOV / 3GP vs FLV / M2TS vs MP4

Related Post of AVI File formats

Related Post of ASF File formats