There are actually a lot of similarities, anonymous. As both are vulpine pokémon, both require the same diet (meat, if not kibble), the same spaces (outdoors to relieve themselves and wide-open areas to run about indoors), and even the same toys (canine chew toys, balls, and soft toys, depending on preference). Moreover, both forms of vulpix share requirements with growlithe, houndour, rockruff, and other dog-like cousins; the difference is that vulpix, being foxes, are a little bit harder to housebreak and leash train. They can be, of course, but it takes a bit more time and patience due to their wilder natures.
The differences between an Alolan and standard vulpix lie in their tolerances to water and heat.
For example, the standard vulpix is partially a fire-type, and like most fire-types, it’s uncomfortable when exposed to high volumes of water. While, like its aforementioned canine cousins, the standard vulpix can be trained to endure water and thus handle a bath, it much prefers to self-groom or have a dry bath consisting of sand and a thorough brushing from its trainer. Additionally, its fire typing means it prefers warmer environments and will often breathe small balls of flame to warm itself up. As such, a trainer needs to be wary of allowing it anywhere near anything flammable, and it’s recommended that they keep their home above 65 degrees Fahrenheit.
By contrast, the Alolan vulpix, being an ice-type, has no problems handling water at all and should be bathed regularly to maintain the cleanliness of its cloud-like tails. However, Alolan vulpix also grow heavily uncomfortable in the heat and thus should be kept cool at all times, even in the summer. If it’s too warm, it may create ice shards around itself to cool itself off, which provide an entirely different domestic hazard for an unwary trainer.
Also, keep in mind that the standard and Alolan vulpix have different types of coats due to the fact that they adapted to different types of environments. The standard vulpix is a short-haired variety that rarely sheds, but it’s very proud of the way its coat shines. It’s necessary, then, for a trainer to brush their vulpix daily to maintain both the smoothness and the shine of this coat; otherwise, the coat becomes dull and matted rather quickly, leading to an irritable vulpix.
Alolan vulpix, meanwhile, are less demanding in terms of personality and more demanding in terms of practicality. As mentioned above, Alolan vulpix have cloud-like tails and manes, which means it’s necessary to brush them to keep their thicker, curlier fur free of dirt and other debris. Additionally, although Alolan vulpix don’t fare well in warmer months, they still have both a summer coat and a winter coat and will shed their fur to cycle through both. It’s not unusual to wake up one autumn morning to find much of your home covered in fluffy, white clouds of fur quite literally everywhere, so invest in a good vacuum cleaner and an equally good pet pokémon brush.
The Pokédex According to Bill 