A big annoyance to atheists is the tendency of religious adherents to claim that atheism is just a different religion. This argument came to prominence by its use by leading religious figures trying to avoid or deflect criticisms of their religion made by prominent atheists (most notably Christopher Hitchens). But the argument itself points out one of the key differences between atheism and religion, namely the existence of choice and options in one's personal philosophy.
I would argue pretty much everyone has a personal philosophy according to which they live their life. The world is a complicated place and without a set of guiding principles and values we wouldn't be able to make sense of it or be able to make decisions effectively. Thus atheist have a philosophy and guiding principles/values just as much as religious people do; the difference between them is who controls them.
Religions are organized sets of beliefs where as atheism (and spirituality) are not. Someone must adhere to religious beliefs, they have no control over them. A religious person is expected to adopt the beliefs taught to them by official teachers (be they priests, rabbis, imams, lamas, or gurus) and seek the guidance of those teachers when ambiguities or contradictions arise; they are not to interpret the beliefs/teachings for themselves. These beliefs or teaching are derived from a small number of officially recognized texts (eg. Bible, Torah, Quran, Shruti). This is to ensure conformity of belief amongst adherents to the particular religion, which is further reinforced by maintaining consistent symbols (eg. cross, star, crescent, Dharama wheel, Aum), ceremonies (eg. Communion, Passover, Ramadan, Holi, Uposatha), buildings/locations (church, synagogue, mosque, temple, monastery), and clothing in conjunction with the beliefs. Various minority religions like Wicca, Sikhism, Jainism, Taoism and mock-religions like Jediism also share these characteristics.
In contrast, atheism is a disorganized mess with the only unifying factor being the lack of belief in deities or the supernatural more generally (ghosts, souls, angels, ...). Each atheist has a personal philosophy of their own choosing be it existentialism, utilitarianism, empiricism, humanism, etc... subject to their own discretion, interpretation, and alteration. As a result 'atheist' organizations are typically named after the particular philosophical beliefs they most closely follow: eg. the Centre for Inquiry follows/promotes skepticism and empiricism, whereas the Secular Humanists follow/promote secularist and humanist ideas.
Spirituality falls somewhere in between being disorganized like atheism but including belief in some kind of supernatural phenomenon (one or more deities, a unifying energy/consciousness, supernatural creatures such as angels, or a soul which persists after death). But importantly spirituality does not have the hierarchical structure or conformity of organized religions which is why many atheists who harshly criticize religion don't have a problem with spirituality.
Many 'religious' people living in Western countries are much closer to believers in spirituality than to 'true' religious believers since they pick and choose which aspects of their favourite religion they want to follow/believe rather than conforming to the official doctrine (eg. 98% of self-described Catholics use contraceptive methods forbidden by church authorities) which is why some people have said the West "believes in believing" in God.
