There are quite a few differences, but for our purposes, only 2 that really matter: gauge (which dictates ohms per ft) and melting point.
Here is a data table I threw together from information found at Kanthal.com:
NIKROTHAL/NICHROME 60 (Nickel, Chromium, Iron Alloy)
675 Ohms Per Circular Mil Ft @ 68F
Max Continuous Operating Temp: 1150°C (2100°F)
Melting Point: 1390°C
GAUGE DIAMETER OHMS/FT
26 0.0159 2.670
27 0.0142 3.348
28 0.0126 4.251
29 0.0113 5.286
30 0.010 6.750
31 0.0089 8.523
32 0.0080 10.55
33 0.0071 13.39
34 0.0063 17.00
Kanthal A1 (Aluminum, Chromium, Iron alloy
872 Ohms Per Circular Mil Ft @ 68F
Max Continuous Operating Temp: 1400°C (2550°F)
Melting Point: 1500°C
GAUGE DIAMETER OHMS/FT
26 0.01590 3.44923
27 0.01420 4.32454
28 0.01260 5.49257
29 0.01130 6.82904
30 0.01000 8.72000
31 0.00890 11.00871
32 0.00800 13.62500
33 0.00710 17.29816
34 0.00630 21.97027
You can essentially get lower ohms with Nichrome 60 using the same gauge wire, but at a cost of a nominally lower melting point (which is what the 'safe zones' in many of the ohms charts are based on). Kanthal A1 is higher in ohms per ft, but has the higher melting point (and can, therefore, handle nominally more power) making it the clear choice if you are building/operating outside the 'normal' range.