Handi or Ultra?
??
Both the same except for the stocks. The Handi has a semi-laminate looking brown wood stock while the Ultra has a Cinnamon true laminated stock.
204 or 22-250?
I can't really say. It depends on what you want it for and if you reload.
The 204 has only a few factory loads and it isn't available as readily as the 22-250.
The ballistics are almost the same except the 250 launches a larger bullet with slightly more energy. They both shoot very flat at long range.
The 204 has bullet weights from 30-50 grains while the 22-250 has weights from 30-80 grains.
The 204 will fire a 32g bullet at ~4200fps~ while the 22-250 will fire a 40g bullet at ~4000fps~. However, a 22-250 will shoot a 30g bullet at about 4500fps, but this will lack due to a terrible ballistic coefficient. Good for close range though!
For larger animals, I would go for the 22-250 but for smaller varmints I would go for the 204. It's all about the usage and if you reload. If you wanted the 204, you would be better suited if you reloaded to stretch its horizons.