spiritual: relating to or affecting the human spirit or soul as opposed to material or physical things.
religious: relating to or believing in a system of religious attitudes, beliefs, and practices.
@zenvelo ‘s answer is good, especially the last two lines.
I however would not use the term “higher power”. Spirituality can also just mean relating to the human spirit, or non-human spirits, or nature or the universe without thinking of it as a higher power. In particular it may involve recognizing your own spirit as on par with all other spirits, or the master of its own experience and views (and possibly in a sense as god-like as anything else), a sense of connection with all things, tuning into intuition, tuning into one’s own consciousness and noticing it isn’t the same thing at all as your material body or your thoughts, etc etc etc all having nothing to do with religion and not necessarily having anything to do with a “higher power”.
And in contrast to what @rockfan wrote about belief in energy or the supernatural, spirituality can also simply be awareness and attention given to one’s experiences or impressions or intuitions or reflections or feelings or consciousness and separating them from material and physical things and ego attachments, and so on, without needing to invoke any sort of “belief” or “supernatural” anything.
And in contrast to what @cookieman wrote, spirituality doesn’t specifically need to be framed as being in conflict with knowing, sensing, or science. Some specific spiritual ideas may be about such things, but not all of them are, and that’s not generally the point of the ones that are.
@rojo No one needs to believe in God to be spiritual. Those Philadelphia Ethical Society guys I would call subscribers to a particular ethical philosophy. The web site says they are a religion but that some of them think of it not as a religion but a philosophy. I would say that there may be some of them who do not identify as spiritual, but that probably on some level most of them are, as valuing the spirits of others probably stems from empathy for the spiritual experience of others, whether they’d choose to use that word for it or not.