Up until Final Fantasy VII, the FF games followed a pattern: odd-numbered releases were less concerned with storytelling and were more technical and gameplay-oriented, while the even-numbered games allowed the player less freedom but offered a more developed cast of characters and strong narratives. FFIII and FFV fit into this pattern perfectly - they're two of the least memorable entries in the series, as far as depth and scope of story go.
That isn't to say the games have poor stories, though. FFIII and FFV actually share a lot of very similar story elements. Both games feature a mostly-unchanging group of party members that simply swaps their jobs as needed. They both feature all-powerful Crystals that control the balance of the worlds and grant the party new job skills.
They do differ significantly, though, in scope and tone. FFIII is based around the traditional "Youngsters go on an exciting quest to save the world" theme. While it's a very easy and enjoyable story to follow, it lacks the sort of emotional punch and sense of urgency you'd find in more recent RPGs. In contrast to FFIII's fairly nondescript kid heroes, FFV's cast is more diverse and interesting, with a pirate, a princess and a grizzled old fighter joining your team. The game's story is also much bigger in its scope, encompassing not just one world, but multiple worlds!
You'll like FFIII if...
You'd like a simple but charming adventure story that doesn't interfere much with gameplay
But you might dig FFV more if...
You like your tales a bit more on the epic side, with multiple worlds and all-out destruction looming in the background
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