Yes, depending on how long you let Kimchi ferment, Kimchi tastes VERY different.
If you like it fresh you can eat it right away. But usually you should let it sit out in the room temperature for couple days (It's processed much faster during summer...and winter it takes little longer and so on) and then store it in the fridge.
Don't freeze Kimchi. It won't taste good so just keep it cool in the fridge and should be able to eat all of it in a reasonable length of time. When Kimchee gets too old/fermented it could be used to cook Kimchi Jjigae! Kimchi Jjigae tastes the best when Kimchi is little too fermented to eat it 'as is.'
When I was 'younger' I used to eat ONLY after it got fermented. I would never eat it fresh. YUCK! so I thought....But as I got older I started to appreciate fresh Kimchi too. I've heard from my parents and other elders that the older you get, the fresher your Kimchi has to be. I guess I am getting old too.
Here's the picture of my mom Young's Kimchi. How does it look?