A suffix tree contains a compact trie like graph representation of all the suffixes of a long string. So if you want to know if it contains a substring, you can start at the root and just search like in a prefix tree. Since for the queried word to be an infix it must be a prefix of a suffix, you will find the word in O(|query|) time.
So yeah its a prefix tree, but a prefix tree of all the suffixes.
And now to answer your question.
If you naively put every suffix into a regular prefix tree, you would get the same lookup performance. BUT, since the suffixes will share a lot of suffixes, which the prefix tree can't compress you have a lot more memory consumption.
The suffix tree avoids this by essentially performing compression on the suffix as well, so that you simply have pointers to shared suffixes of suffixes, which point to the root.
In the end it's like a diamond shape I guess, trie in the front, compression in the back.
The underlying data structure (a trie[0]) is used to create both suffix trees and prefix trees. The 'default' trie is a prefix tree actually.
The reason you would want a suffix tree (over a prefix trie) is that suffix trees are quite useful for computing the longest common substring of two strings (they can do it in linear time).
Peter Weiner, who first invented suffix trees gave an answer to this question on p.3 of "Linear Pattern Matching Algorithms"[1]: "If P does match a substring of S, then reverse(P) also matches a substring of reverse(S). This observation implies that every technique which solves a pattern matching problem working from left to right has a dual procedure which works from right to left. In what follows, we adopt a left to right view point, referring only briefly to dual concepts as appropriate."
The key property is that each substring is the begining of a suffix, so if you have your suffixes ordered in lexicographical order, is easy to answer pattern matching queries. Now, you could do the same with prefixes, but then you would need to reverse the query pattern, and things become less straightforward.
The prefix tree of a string is just a linked list, though—it's the string itself. It doesn't help you do anything for a single string, unlike a suffix tree, which is used for finding substrings of a single string.
Moreover, how would you even start searching for a substring using a prefix tree? You can't start at the root, because the substring wouldn't be a prefix. It doesn't help you there at all.
For most problems, a prefix tree will be just as useful.
But suppose you want to look for the best match of string in a data stream with no predetermined length e.g. a TCP pipe. Then a suffix tree will allow you to process each incoming character in constant time. A prefix tree will not.
If you have keys that tend to have common prefices (e.g. URLs in RDF) then checking them in reverse (e.g. with a suffix tree) is generally faster, as you will hit differences quicker.
