Is any mathematican more famous for their conjecture(s) than their theorem(s)? 
I'm wondering if some mathematicians gained more fame from their (perhaps visionary) conjectures, than from the positive results they proved?

I would say this is not true of Fermat, despite his famous eponymous conjecture
(now settled),
because he established so many results independent of his conjecture.
And this seems not true of Poincaré, whose famous conjecture (also now settled) bears his name.
But he was incredibly accomplished independent of that conjecture.
Atiyah has formulated wonderfully productive conjectures (one leading to a Witten advance), but he has also
established major results, e.g., the Atiyah-Singer theorem.
I am interested to explore whether some mathematicians have specific conjecture-talent that is not evidently reflected in theorem-proving talent.
 A: Taniyama-Shimura-Weil conjecture which states that elliptic curves over the field of rational numbers are related to modular forms. Andrew Wiles used this conjecture to establish the modularity theorem for semistable elliptic curve. This became the basis of Wiles proof of Fermat's last theorem.  Yutaka Taniyama never lived to see the fruits of his work as he had committed suicide at a young age. Taniyama once remarked that he himself did not fully understand the conjecture and many mathematicians thought that it was a tough conjecture to work upon. 
A: Mikhail Yakovlevich Suslin made some important discoveries, and no doubt would have made many more had he not died tragically young, but he is most famous for Suslin's problem. It is true that he stated it as a question rather than a conjecture, but I'm not sure if you are insisting on that distinction. Rightly or wrongly it's often called Suslin's hypothesis; the difference between a hypothesis and a conjecture seems kind of nebulous to me.
A: Most of the people who have famous conjectures would be pretty well-known even without those conjectures, but maybe these are some exceptions (from Number Theory): Giuga; Catalan; Bertrand; Waring. 
A: Bernhard Riemann.
Riemann died shortly before his $40$th birthday.
Lots of things are named after him:
see here.
However, the most famous one is probably the Riemann hypothesis.
A: Lothar Collatz: while he is a celebrated mathematician who has a formula named after him (Collatz-Wielandt formula) and he has received quite a few honorary degrees (see wiki), he is definitely best known for his Collatz conjecture, also known as the $3n+1$ conjecture, which he posed in 1937.
It remains unsolved up to this day, despite numerous attempt by professional and amateur mathematicians, and its popularity can be seen even here. In fact, I would argue that the best "proof" that his conjecture is more famous than his actual work, is that the tag (collatz) refers exclusively to the Collatz conjecture.
A: Andrew Beal, while not strictly a mathematician, has not proven (to my knowledge) any mathematical result, despite having a rather famous unsolved problem in his name with a monetary prize that is the same of that of any Clay Mathematics prize. 
A: Franz Mertens was a late 19th/early 20th century German mathematician who is known for some results about the density of prime numbers and even has a constant named after him. But what he is probably most famous for, no doubt because of the elementary nature of the statement, is the Mertens conjecture, which stated:

Mertens' conjecture: For all $n>1,$ $$\left\lvert \sum_{k=1}^{n}\mu(k) \right\rvert < \sqrt{n},$$ where $\mu\colon\mathbb{N}\to\{-1,0,1\}$ is the Möbius function.

This conjecture, if true, would imply the Riemann hypothesis (!) - hence its fame. Unfortunately, in spite of tremendous amounts of numerical evidence in favour, it was proved false in the 1980s. The smallest counterexample is known to be larger than $10^{14},$ but the upper bound on it is truly enormous: $\operatorname{exp}{(1.59\times10^{40})}.$
A: As others have pointed out, "famous" is subjective and a bit ephemeral.  I would like to adjust the question to ask for mathematicians who have actually contributed more to mathematics by their conjectures than via their theorems.  This is still subjective, but distinctly less so.  
Anyway, when I try to think of a mathematician whose greatest contributions lie with his conjectures and for which those contributions are enormous, one name springs to mind: Robert Langlands.  The Langlands program is one of the most important and influential pieces of 20th and 21st century mathematics.  I would imagine that Langlands himself would agree that his program is (even) more important than the results he has proved: in fact, most or all of his important results feed into and elevate his program.  
The OP says

I am interested to explore whether some mathematicians have specific conjecture-talent that is not evidently reflected in theorem-proving talent.

I think this is a good example of this, of a certain particular kind: Langlands clearly has remarkable theorem-proving talent.  However his conjecture-talent is beyond remarkable...it is Langlandsesque.  
I think there is a further small perturbation of the question which makes the canonical answer Paul Erdős. If we rank mathematicians by number of theorems proved then Erdős surely comes near the top of the list.  However, the influence that he has had on contemporary mathematics goes beyond any one result of his.  Erdős died (almost exactly) 20 years ago.  That was right about the point where I started paying attention to the mathematical landscape, in particular number theory.  The last 20 years have been a stampede towards the kind of problems that Erdős proposed.  In particular, the amount of leading mathematical work done in that time towards the Erdős–Turán conjecture alone is enormous. 
