"On the map there are as many rivers as mountains."

Translation: A térképen annyi folyó van, ahány hegy.

[BryanZFC]

What is the difference between "ahány" and "amennyi"?

[pythagoras322]

When you say:
"There are as many/much X as Y" your sentence will look like this:
"Annyi X van, amennyi/ahány Y"

So at the first part you always use "annyi" and at the second you use "amennyi" or "ahány" according to the noun(s) you are talking about.

Basically "amennyi" = "as much" ; "ahány" = as many"

So following this logic:
"On the map there are AS MANY rivers as mountains." = "A térképen annyi folyó van, AHÁNY hegy."
( Note that the first part always takes "annyi")

Anyway the "amennyi" or "ahány" problem is basically the same as the "mennyi?" or "hány?" problem.
"Mennyi?" = "How much?" ; "Hány?" = "How many?"