Q:

Why is stroke order important?

While I understand that stroke order is important, I do not understand why it is important. If the final product would look the same, why does it matter? I don't have an issue following stroke order. I'm just curious.

A:

Stroke order is important for hand-written Japanese, which includes normal handwriting and various styles of calligraphy.
The stroke order gives a flow to the character that can be recognized, even when the character looks very different to its [楷書]{かいしょ} incarnation.
For the non-expert, a character written in 楷書 (in the correct order) probably cannot be distinguished from the same character with a different stroke order. But as soon as you get into cursive styles, the difference of the same character with different stroke orders becomes very obvious.

If the author of the [玉泉帖]{ぎょくせんじょう} above had written like he did, but with a random stroke order, probably nobody would be able to actually read it.

A:

When you want to ask a Japanese person about a kanji/word... they may ask you to write it out. If you trace out the character with a finger on your palm IN THE CORRECT ORDER, they will probably be able to recognize the strokes and answer your question quickly. This shows up way more often than you'd expect.
Frankly, it's easier to remember complex kanji if you basically understand stroke orders. I'm not saying it's essential, but it does help your mind break them down a bit easier.
If you ever want to read highly-stylized characters (particularly sake brand names and store signs), or handwriting, you'll need to know stroke orders to help decode it. Kanji-shorthand is extremely difficult to parse if you're not sure what order it could possibly have been put together in.
If you ever have to write something (although this is probably going to be fairly rare),  the stroke order is important to legibility in many cases... although this is probably the least important reason to learn them.

With that said... for alot of radicals it doesn't make that much difference. 左右's first two strokes are in opposite orders, and I doubt many people would notice. Stroke order is also generally not that hard to learn. Since after about 100 or so basic radicals (most of the bushu radicals are made up of simpler ones), you don't have that many exceptions left that will trip you up.

