They are very similar.
A wizard who wished to create a Horcrux would then use that damage to their advantage by casting a spell which would sever the damaged portion of the soul and encase it in an object.
...their soul and vitality were successfully stored in the jar...
So a phylactery contains the lich's entire soul1 and its vitality where a horcrux contains a damaged sliver of the soul.
Their effects are slightly different as well. As noted in your question a lich is undead as a necessary component in the phylactery's creation. Voldemort was still alive after the creation of his horcruxes.
A lich whose physical body is destroyed pops back up after a few days near where their phylactery is hidden. They may also possess a dead body in the phylactery's vicinity.
If a lich's physical body was destroyed, it could regenerate itself where its phylactery was located within a tenday, with a new body appearing adjacent to it.
...reemerge and possess any dead body in the vicinity, whether their own, another's...
Voldemort was trapped in an incorporeal form.2 He could latch onto a willing person, as he did with Quirrell, or a ritual to regenerate himself.
flesh - of an amount about as large as a person's hand - willingly sacrificed from one of his servants
Someone using a horcrux seems to be less self-sufficent than a lich as they need willing participants to fully come back from having their body destroyed.
A lich must feed souls to the phylactery to keep it sustained or else they fade away or become a demilich. There is no indication that applies to horcruxes.
To sustain the power of the phylactery and their undead existence, a lich needed to fuel it with the souls of others.
In summary they are basically the same, unless you're really diving into the details of how a lich operates they are functionally the same thing.
1 There is a spell that can be used to create more phylacteries, presumably that would mean a portion is in each.
2 I'm unsure if that is inherent to horcruxes or if Voldemort is a special case due to the damage he did by creating so many.