By real-world standards, if it has complex thoughts, moves under its own power, produces sounds via vocalization - especially speech - and has a full array of emotions, it must be an animal, and since humans are animals, Ents are more like real-world humans.
But in Tolkien's world, they are clearly more like trees than they are like people. Ents can become "treeish" and trees can become "Entish", for one thing. An Ent that has become thoroughly treeish is basically a normal tree - it doesn't walk or speak as often, if ever. And unlike trees in our world, at least some trees in Tolkien's world can do things like move around, feel emotions (they are frequently described as hostile, resenting things that go about on legs or wings), move around a bit (like the Huorns, who seem to be trees that have become entish, or Old Man Willow, who attacked the hobbits and may have been a Huorn). Treebeard says that he knew many of the trees - and he calls them trees - that were cut down by Saruman's Orcs, and he refers to them as "friends" who "had voices of their own".
What does all of this mean? It means that, whereas in our world the difference between trees and animals is pretty obvious, in Tolkien's world, the boundary is much harder to define, if it exists at all. Ents are clearly "more than trees", but the difference is one of degree rather than essence. In other words, Ents are more sentient than trees, and a very treeish tree is not an Ent, but as someone mentioned earlier, an Ent is basically a tree PLUS something else. The difference between trees and animals in Tolkien's world is very vague; the difference between trees and Ents is almost nonexistent.
The most we can say with certainty is that Ents are related to trees, and resemble trees to a great extent: they have bark (or something like it); they seem to drink but not eat; they have branches and root-like toes; they can become "treeish" and essentially turn into trees. Still, they act a lot like people and other humanoid beings (Elves, Wizards, etc): they walk around; they talk; they think; they have legs, a face, arms, heads, hands, feet, and even toes; they form relationships with trees, other Ents, humanoids, and possibly non-humanoid animals; they have complex emotions; they have "wives"; and so on.
But most of the things they have in common with humans are not unique to humans, or even to animals, in Tolkien's world. At least since the Elves "woke" some of the trees and taught them to speak, many normal trees have been able to think, move, speak, and feel emotions, perhaps not as fully as Ents or animals can, but to some degree at least.
So the answer is a complicated one: By our standards, Ents are more like people than trees, because they can do things trees in our world can't. But our standards don't apply to Middle-earth, and by Middle-earth standards, Ents are more like trees than anything else. They aren't trees, or at least they didn't start out as trees, but they can become something that is essentially indistinguishable from a tree.
In a world that follows the laws of magic rather than science, and where mythology has not yet been replaced by history, this question probably doesn't make much sense. An Ent is an Ent, a tree is a tree, a person is a person, and the boundaries between them are vague at best, perhaps even nonexistent. No one in Middle-earth seems to think about these things much - like the relationship between humans and hobbits, for example. Hobbits see humans ("Men" or "Big People") as a separate species, but there is reason to believe that hobbits might be a type of human. This is an interesting question to us, but apparently not to hobbits and humans in Middle-earth.