These terms are popularly used quite interchangeably, but there are distinctions one can make, which are clearest in the rules if heraldry.
A patch is a piece of cloth sewn onto a garment, presumably in this sense this cloth is embroidered with a symbol, which one frequently sees on military or sports uniforms. The symbol/emblem could come from many sources,such as a company logo, a tribal symbol, a heraldic charge (from a coat of arms) , or from the crest that sits above the shield in a coat of arms. An example can been seen at
https://thegenegenieblog.wordpress.com/2017/07/09/coat-of-arms-and-crests-whats-the-difference.
The crest originally was an object placed on top of a knight's helmet, which might have been inspired by the symbols on his coat of arms, but which could also be something different. Animals as crests, for example were often modelled from painted boiled leather! The use of the crest alone as a symbol is illustrated at
http://www.gg.ca/document.aspx?id=78. This example also shows how different the crest is from the overall coat of arms, which is why the popular use of "crest" is inaccurate.
As my own user tag suggests, I have an interest in heraldry and its modern use in many areas, including sports.
