To keep it simple for those learning Spanish, as a native speaker I would translate the verbs as follows: Amar = to love, used between loving partners, like a husband wife combination. I would never tell a close friend of mine, or family member, for example, te amo. However, in some spanish countries it might be normal to say te amo to a close family member, specially your mom. In fact, little kids might tell their moms, "te amo," probably because at an earlier age, it is easier to pronounce than "te quiero." But as you get older, you start to tell your parents, "te quiero," "los quiero." But, yes, I've heard some friends of mine tell "te amo" to their mom, specially. And as a kid, you have a saying, "amo a mi mamá." (I love my mom). Querer = to want, to love. Querer can be used to express "i love you" to friends, family members, even your pets, if you so desire. Also, if you're starting a relationship with someone, usually you would tell the person, te quiero and not te amo. An easy comparison would be for example, if you meet a person today, and you like that person; you go on a 2nd date, and a 3rd date. You might call your friend and say, "I'm going on my third date because I like this person." It would be strange to say, I'm going on a third date because I love that person. Love carries a deeper feeling than liking. That's kind of the difference between querer and amar, as you are dating somebody. And yes, querer can also be used between loving partners to express love, and I know many couples who would say in public te quiero, but they wouldn't say te amo because it kind of sounds dramatic---well, at least in public. Te amo is something you would tell your loved one in a private setting, unless you're out with close friends and the other couple is comfortable in that setting using the word amar. Querer also means to want, as in, quiero una galleta (I want a cookie). Quiero ir a la biblioteca (I want to go to the library), etc.

"Amar" and "Querer" are two different words that has technically the same meaning: "To love"; but the context in which it is used is different; but sometimes there is a blurry line between the context. Traditionally when you say "Amar", you usually use it when you , are in a romantic relationship but also, it can be used in a father/mother-son/daughter relationship. What I mean is that, the word "Amar", tends to be more formal. In contrast, when you use "Querer" the meaning tends to be more lighter and usually refers to the feeling between friends or a person that one is close to. But, there is a blurry line between them, in the sense that, both words: "Amar" and "Querer" often are interchange; it can be used to portray the same meaning:"To Love". And often, they are used that way in Spanish. But, if we think in a strict sense: The word "Amar" will refer to a STRONG feeling and "Querer" will refer to a LIGHTER feeling. This word (LOVE), is specially interesting in Spanish, because in this language there is always a different level in terms of the feeling that the word portrays. For example, in Spanish you could say: "Te quiero", "Te amo", "Te aprecio", "Te adoro", etc.; and each word has a different and subtle meaning, depending on the person and the situation that the word is used on.

Im not a native speaker but from what I understand you would only say "te amo" in a romantic relationship and you would use "te quiero" for friends and family that you love. Again im not sure if this is exactly correct but Ive heard that before