iPhone 5S: 1.2 megapixels at 1280-by-960-pixel resolution. Backside illumination sensor.
Lumia 1020: 1.2 megapixels at 1280-by-960-pixel resolution. f/2.4 aperture.
iPhone 5S: gold, gray, silver
Lumia 1020: yellow, white, black
iPhone 5S: Uses iTunes Store for downloadable music; no terrestrial radio support. Supports AAC, Protected AAC, HE-AAC, MP3, MP3 VBR, Audible (formats 2, 3, 4, Audible Enhanced Audio, AAX, and AAX+), Apple Lossless, AIFF, and WAV file formats for playback.
Lumia 1020: Uses Xbox Music or Nokia Music for streaming music; includes FM radio. Supports MP3, AMR-NB, WMA 10 Pro, GSM FR, WMA 9, AAC LC, AAC+/HEAAC and eAAC+/HEAACv2 codecs. Also supports ASF, MP4, AAC, AMR, MP3, M4A, WMA, 3GP and 3G2 file formats for playback.
iPhone 5S: Records at up to 1080p (Full HD, 1920-by-1080-pixel resolution) at 60 frames per second. 3X video zoom. Supports H.264, MPEG-4, M-JPEG, as well as FaceTime video calling with similarly enabled devices; AirPlay mirroring and video out to Apple TV, video mirroring with adapter. Includes slow-motion video capability, video stabilization, face detection, geotagging and ability to take still photo while recording.
Lumia 1020: Records at 1080p (Full HD, 1920-by-1080-pixel resolution) at 30 frames per second. 6X video zoom. Supports VC-1, Windows video, H.264/AVC, H.263 and MPEG-4 codecs as well as MP4, WMV, AVI, 3GP, 3G2, M4V and MOV file formats for playback. Includes video streaming, five white balance modes, video stabilization, autofocus.
iPhone 5S: Unknown
Lumia 1020: 2 gigabytes
iPhone 5S: 16, 32 or 64 gigabytes local; 5 gigabytes free in the cloud
Lumia 1020: 32 gigabytes local; 7 gigabytes free in the cloud
iPhone 5S: Fingerprint sensor, slow-motion video capture, OpenGL ES version 3.0 support for higher-res gaming, free iWork apps.
Lumia 1020: That 41-megapixel camera, which allows you to zoom to crop down to the photo you want; an extra-sensitive touch-enabled display that works with gloves; free Microsoft Office apps.
This list can't be everything to everyone, but if you're interested in more specifications or features that are not included here, visit the detailed product pages for the iPhone 5S and the Lumia 1020. If we've missed anything major, just give us a shout.