Friend is the term used in AmE meaning someone with whom you have a bond.
Mate is the term used in BrE to refer to a friend (in the AmE sense) but may also be a cordial way of referring to another person whom you may or may not know.
These two phrases are equivalent in their respective meaning of someone close
In AmE, one's mate is usually though of as one's significant other or romantic partner.
These phrases are equivalent ways of greeting someone:
Hey mate (BrE)
G'day mate (AusE)
In BrE, "Hey mate!" can be used as a general greeting (usually amoung males), however
"Hello friend!" is usually not used in AmE, more often "Hi there!" might be used.
Without any context, if a boy and a girl are said to be friendly
In AmE it may be thought they are friends or cordial
In BrE it may be thought they are more than just friends, they are friends with benefits (AmE)
One would never say they are matey.
Other BrE words which convey friendship are pal and chum.
Pal is more often used in Scotland than mate but is interchangeable in meaning.
When two people are chummy, they are very close friends.
Chums was a parody of the AmE series Friends.