Why does the curvature of space decrease during inflation but increase after it? According to the Friedmann equation, curvature of space will increase with time/expansion of space, but I've also read that during Inflation, the expansion caused the Universe to flatten out. What is the difference between these two types of expansions?
 A: According to the Friedmann equation, as space expands, then if it has curvature, that curvature decreases as the scale factor increases. So I don't think there is any contradiction with what might happen during inflation.
The main difference with the inflationary epoch is that the radius of curvature increases exponentially with time.
A: 
Why does the curvature of space decrease during inflation but increase after it?

There are some issues here. One is that curved spacetime isn't the same as curved space. Another is that we don't actually know that space is curved or whether this curvature has decreased. And on top of that we don't actually know that inflation actually happened. Have a read of Physicist Slams Cosmic Theory He Helped Conceive: "In subsequent years I talked to Steinhardt about cosmology and in particular inflation, an idea that he helped refine in the early 1980s. Inflation holds that immediately after the big bang, our universe underwent an almost unimaginably explosive, faster-than-light growth spurt. Lately, Steinhardt has been criticizing inflation and related ideas, notably multiverses, in unusually blunt terms. So I was delighted when he agreed to answer some questions."

According to the Friedmann equation, curvature of space will increase with time/expansion of space

Only WMAP indicates that space is flat: "Recent measurements (c. 2001) by a number of ground-based and balloon-based experiments, including MAT/TOCO, Boomerang, Maxima, and DASI, have shown that the brightest spots are about 1 degree across. Thus the universe was known to be flat to within about 15% accuracy prior to the WMAP results. WMAP has confirmed this result with very high accuracy and precision. We now know (as of 2013) that the universe is flat with only a 0.4% margin of error."
So the curvature of space can't have been increasing. It's like what Rob said, if it has curvature, that curvature decreases. And note that two out of the three "shapes" of the universe were always going to be wrong. To me that doesn't inspire confidence. IMHO the Friedman equation should predict what we're going to see, not cover all the bases.
 Public domain NASA image

but I've also read that during Inflation, the expansion caused the Universe to flatten out. What is the difference between these two types of expansions?

One is hypothetical, proposed to solve the flatness problem. The other is factual, supported by hard scientific evidence. IMHO when you go back and read the Einstein digital papers and note where he described a gravitational field as space that was neither homogeneous nor isotropic, and then note that the Friedmann equations concern the expansion of space in homogeneous and isotropic models the flatness problem goes away. Space is flat, it was flat a billion years ago, and a billion years before that. It's always been flat, because on the large scale space is homogeneous so light goes straight. It's expanding, but inflation is superfluous.
