One of the most obvious differences are that ascended beings are animals while aspects still have a human form. But who is more powerful and what relationships do they have with each other?
The Ascended are people who gain power from a ritual involving the Sun Disc. Since this is explicitly called something related to the stars and celestials it is implied that they’re using space magic in the transformation. This makes them bigger stronger more durable and can heal from wounds that would kill a Mortal. It may also increase their magic proclivities.
The Aspects are a race of beings that live on some other realm dimension of existence. Their true powers are vague and unexplored but they are said by Riot to be comparable in strength to Aurelion Sol.
The Aspects interact on the plane of existence that people are on through hosts that they partially or fully possess. These agents have a wide range of abilities. Some can teleport across dimensions, harness the power of stars, and can stop armies from fighting each other. While others... fight good. Apparently.
the ascended are actually well-written and 3-dimensional characters. The aspects are all horrible people.
The Ascended seem to have powers based on controlling sand and other curse like effect; Nasus' wither and Azir essentially recreating images of the dead. This is brought about through channeling the power of the Sun into a warrior, mimicking the Targonian's ritual of binding with a host.
The Aspects are persons chosen by Ancient Targonians to host their celestial bodies, in order for the Targonians to have influence on the planet, in this case Runeterra. Being celestials means that the magic the hosts receive are celestial in nature, hence Leona and Diana's Sun and Moon powers, Zoe's Time-Space warping and manipulation of stardust, Taric's gems are actually condensed starlight and Pantheon has been shown to be able to withstand a certain degree of Starfire (whether or not this was because it was his own magic isn't confirmed). This plus super human, strength, durability and agility in Pantheon's case at least.
