[CRGreathouse]

Using (to avoid the extra variability from the small primes that brings) and searching an interval around 10²⁰ I find

1 distinct prime factor: 4290
2 distinct prime factors: 21379
3 distinct prime factors: 44810
4 distinct prime factors: 54544
5 distinct prime factors: 42306
6 distinct prime factors: 22179
7 distinct prime factors: 8090
8 distinct prime factors: 2022
9 distinct prime factors: 331
10 distinct prime factors: 49
11 distinct prime factors: 1

which compares to the (naive) Landau predictions of

1 distinct prime factor: 4342
2 distinct prime factors: 16632
3 distinct prime factors: 31849
4 distinct prime factors: 40658
5 distinct prime factors: 38928
6 distinct prime factors: 29817
7 distinct prime factors: 19032
8 distinct prime factors: 10412
9 distinct prime factors: 4984
10 distinct prime factors: 2121
11 distinct prime factors: 812

and the Erdős-Kac predictions

1 distinct prime factor: 29250
2 distinct prime factors: 51228
3 distinct prime factors: 70509
4 distinct prime factors: 76319
5 distinct prime factors: 64973
6 distinct prime factors: 43492
7 distinct prime factors: 22872
8 distinct prime factors: 9438
9 distinct prime factors: 3051
10 distinct prime factors: 772
11 distinct prime factors: 152

This supports the intuition that Landau is better for small numbers of prime factors and Erdős-Kac better for large. In this case the crossover is surprisingly large (8 prime factors) but neither estimate is particularly accurate.