To use ffmpeg programmatically, I think you would have to call read_apic() in libavformat (which is part of ffmpeg).
From the commandline, you can apparently do this:
ffmpeg -i input.mp3 -an -vcodec copy cover.jpg
The commandline behaviour implies that the cover art image is seen as just another video stream (containing just one frame), so using libavformat in the usual way you would to demux the video part of a stream should produce that image.
Sample code for demuxing: ffmpeg/docs/examples/demuxing.c The first (and only) AVPacket that would be obtained from demuxing the video stream in an mp3 would contain the JPEG file (still encoded as JPEG, not decoded).
AVFormatContext* fmt_ctx;
// set up fmt_ctx to read first video stream
AVPacket pkt;
av_read_frame(fmt_ctx, &pkt);
FILE* image_file = fopen("image.jpg", "wb");
int result = fwrite(pkt.data, pkt.size, 1, image_file);
fclose(image_file);
If there are multiple images, I think they would be seen as separate video streams, rather than as separate packets in the same stream. The first stream would be the one with the largest resolution.
All this is probably implemented internally in terms of read_apic().
The ID3v2 spec allows for any image format, but recommends JPEG or PNG. In practice all images in ID3 are JPEG.
EDIT: Moved some of the less useful bits to postscript:
P.S. ffmpeg -i input.mp3 -f ffmetadata metadata.txt
will produce an ini-like file containing the metadata, but the image is not even referred to in there, so that is not a useful approach.
P.S. There may be multiple images in an ID3v2 tag. You may have to handle the case when there is more than one image or more than one type of image present.
P.S. ffmpeg is probably not the best software for this. Use id3lib, TagLib, or one of the other implementations of ID3. These can be used either as libraries (callable from the language of your choice) or as commandline utilities. There is sample C++ code for TagLib here: How do I use TagLib to read/write coverart in different audio formats? and for id3lib here: How to get album art from audio files using id3lib.