There are differences in law between civil partnership and civil marriage. A civil partnership is formed when the second partner signs the Civil Partnership Schedule. A civil marriage is formed when the couples have exchanged their declaratory and contracting vows.
Opposite sex couples can opt for a religious or civil marriage ceremony as they choose, whereas formation of a civil partnership will be an exclusively civil procedure. There are also differences in the content of a marriage and a civil partnership ceremony.
Same sex couples can be married in religious institutions if the religious institution specifically opt in to the Marriage (Same Sex Couples) Act 2013 to do so, with written authorisation from their governing authority to allow for a place of worship to be registered for same sex marriages. It is a matter for the individual religious organisations whether they wish to do so.
A comparison between civil partnership and same sex marriage can be found on the GOV.UK website.