Internet Explorer and Edge do not support some MP4 formats that Chrome does. You can use ffprobe to see the exact MP4 format. In my case I have these two videos:
Input #0, mov,mp4,m4a,3gp,3g2,mj2, from 'a.mp4':
  Metadata:
    major_brand     : isom
    minor_version   : 512
    compatible_brands: isomiso2avc1mp41
    encoder         : Lavf56.40.101
  Duration: 00:00:12.10, start: 0.000000, bitrate: 287 kb/s
    Stream #0:0(und): Video: h264 (High 4:4:4 Predictive) (avc1 / 0x31637661), yuv444p, 1000x1000 [SAR 1:1 DAR 1:1], 281 kb/s, 60 fps, 60 tbr, 15360 tbn, 120 tbc (default)
    Metadata:
      handler_name    : VideoHandler
Input #0, mov,mp4,m4a,3gp,3g2,mj2, from 'b.mp4':
  Metadata:
    major_brand     : isom
    minor_version   : 512
    compatible_brands: isomiso2avc1mp41
    encoder         : Lavf57.66.102
  Duration: 00:00:33.83, start: 0.000000, bitrate: 505 kb/s
    Stream #0:0(und): Video: h264 (Constrained Baseline) (avc1 / 0x31637661), yuv420p, 1280x680, 504 kb/s, 30 fps, 30 tbr, 15360 tbn, 60 tbc (default)
    Metadata:
      handler_name    : VideoHandler
Both play fine in Chrome, but the first one fails in IE and Edge. The problem is that IE and Edge don't support yuv444. You can convert to a shittier colourspace like this:
ffmpeg -i input.mp4 -pix_fmt yuv420p output.mp4