A radix tree is a compressed version of a trie. In a trie, on each edge you write a single letter, while in a PATRICIA tree (or radix tree) you store whole words.
Now, assume you have the words hello, hat and have. To store them in a trie, it would look like:
e - l - l - o
/
h - a - t
\
v - e
And you need nine nodes. I have placed the letters in the nodes, but in fact they label the edges.
In a radix tree, you will have:
*
/
(ello)
/
* - h - * -(a) - * - (t) - *
\
(ve)
\
*
and you need only five nodes. In the picture above nodes are the asterisks.
So, overall, a radix tree takes less memory, but it is harder to implement. Otherwise the use case of both is pretty much the same.