Biologically there is no doubt about it. Humans are animals. We have evolved from the same ancestors with the primates, our ancestors had common ancestors with the canines, the felines, the avians, they had common ancestors with the aquatic life, and the list goes on till the point of major evolution branches, where the important changes happen, thus setting animals, plants, fungi, bacteria and parasites apart. When you share 99% of your DNA with a chimpanzee, you cannot claim to be much different than it, at least biologically.
Do you wish for further proof?
In order to sort humans as a different lifeform, you will need physiological and anatomical criteria. These begin from the most superficial examination of the skin, the most basic organ systems and the organs themselves, all the way towards tissues and cells. We can go even further to examine the organelles, the proteins and even the DNA, but that's reserved for minor differences between already verified evolutionary-close species (usually residing under the same or neighboring genus). So, if you check the organs and tissues between a human and a plant you will find noticable differences. However, between humans and other animals, usually the same things appear; a heart, a digestive track, a blood maintainance network, a central nervous system, and a movement aparatus. That's basically all you have on your body, save for the reproductive organs, and your air filter.
Do you need us to go deeper?
Challenge accepted. Tissue level.
Even if the proteins you create for specific tasks are much different to those of other animals, usually the basic structure is the same, resulting to similar shape of cells in the respective tissues, thus creating patterns that are close to being exactly the same. That does not necessarily have to do only between close species like a dog and a wolf, but it's also that common that you share 55-65% of your tissue structure with Lab Rats, and over 75% with Dolphins. If you decide to kill the poor things, take out their flesh and examine their sceleton, you will find the same structure, even in their extermities, with the exception of the tail, which for humans would be an evolution residue. We used to get insulin from pig pancreas before we could modify bacteria to create it for us. This means we had so similar metabolism to those pigs our digestive system is macroscopically the same. Check how many rats, bunnies, and monkeys are used as lab-experiments for future medication and ask yourself why that happens. If you still have questions re-read this reply, and if it's not enough, open an anatomy book.
We could go even deeper as to the mechanisms in gene transcription, that are so similar among animals, it's a joke we are even having this 'debate'. The fact that our brain is more evolved makes us no different. Others have invested in survival mechanisms as bigger claws, feet, longer tails, extended skin for flying or feathers, kidneys to survive in harsh heat, or skin for the cold, lungs to last longer, or antlers to hit from a safe distance. In the chaotic way matter works, it's not wonder a species evolved with a highly functioning nervous system. Remind you, the renal cortex is that common, that even a fish has it. So even if our claim of superiority is correct, we are also limited.
From a psychological point of view, humans are similar to other animals in almost everything. Our insticts, our needs, and our habits are exactly the same. Even though some animals prefer different lifestyles, because of the area they evolved in and the dangers it hid, most animals are social creatures that either travel or hunt in packs. However, do not confuse the modern lifestyle with your natural affinity for your most primal abilities and instincts. The emotional bond of the baby with its mother in its early stages is as vital as proper nutrition. When the first is absent, substitutes are necessary. Intuitively you might not grasp this at first glance, but it you notice the behavioral patterns of elephants, cats and birds, you'll notice the resemblance. In the first case the substitute is another elephant due to the long gestation cycles of those beasts, and in the second case you'll see the siblings act as substitute, and if that case is eliminated fast, the baby will resort to an object of its choice. If you have seen a dog move around its favorite toy and/or not be able to sleep without it, I suggest you also check all the human babies that do the same.
The only reasons one would separate us from animals as far as our mental abilities and psychology is concerned are the following 2
a. Verbal communication and expression of feelings with it.
b. Higher consciousness, aka consciousness of our consciousness.
The first is more obvious than the second. All animals produce sounds and based on their frequency and timbre, they send each other messages. That's merely 10% of how every animal communicates, as over 90% of our communication is non-verbal, and that's why people pay attention to body language, facial expressions, possibly smell (even though humans have long lost this ability due to our reduced smell over the centuries), sounds from various body parts etc. As a human, you express most of your messages in this way too. However, two regions on your cerebrum have evolved specifically to a. understand patterns of sounds or symbols as speech, and b. co-ordinate your thoughts, vocal cords, mouth, and thorax into creating speech with your mouth, or your upper extremities into creating written speech. That is unique in the animal kingdom, even if those areas are small compared to the rest of your brain (Wernicke and Brocka for those interested). Speech is integrated part of your biology and defines your communication with other animals, without limiting that to humans. You can train certain animals with more evolved cortex to communicate with you, even if their abilities are limited. A dog can answer to commands, a parrot can replicate your speech, an ape can be trained to do complex exercises (fun fact: it has a better 'RAM' than you). You are ahead of them because of evolution.
The second is the ability to understand abstract thoughs and ideas, elaborate on them, apply meaning to them, and learn from them. When you see a flag, you apply meaning of nationality, history, culture, pride, and stereotypes on it. When a dog sees it, it's just a piece of cloth. It has no meaning for it, and sometimes it can't even tell all the colours on it (don't worry, dogs will know if it's an American or Australian flag, however not a Brazilian one). This way humans will contemplate the meaning of happiness, love, legacy, friendship, motherhood. The dog can also understand friendship, but it cannot ponder at its general meaning, the way it affects the relatonship with other people, animals, or objects, and it will never know what it really means when you ask "who's a good dog?". Not only because it cannot understand what you are saying, but also because it cannot understand the concepts of good and evil, unless they directly harm it or someone it cares about, or affect it in other ways it will feel limited, free, or otherwise mending with its basic insticts. After all that's all we end up talking about.
These are the things you see animals around you and probably think you are superior to them. Up to a certain point, you are correct. The spider looks at the fly and knows it has full power over its life. Do not think you are much different. Biologically or Psychologically, your basis is the same, and if I were to post about DNA, you wouldn't even get a chance of rebuttal. A debate or dillemma has to have another side. If you create one just for the sake of it, at least pay some respect to centuries of proof that's coming towards you, and the people that work to preserve said evidence.