grammar - Difference between「と思います」and「かと思います」?

What is the difference in meaning and usage between「と思います」and「かと思います」at the end of a sentence?

Answer

There's a difference in meaning between 〜かと思う and 〜と思う.

Rhetorical usage:

心臓が止まるかと思った
"I thought my heart was going to stop!"

心臓が止まると思った
"I thought my heart would stop."

(I changed the tense in the English, but ignore that. The difference is in tone.)

心臓が止まると思った just means that you had the thought that your heart was going to stop. 心臓が止まるかと思った on the other hand is rhetorical. 死ぬかと思った is another example of this usage.

Softening usage:

そうする意味はないかと思います。
"I think there's probably no significance in doing that."

そうする意味はないと思います。

"I think there's no significance in doing that."

The bottom one here is making a strong statement (despite the 〜と思います, it's basically just declaring what you think). The top makes more sense if you're talking in a context where it is better to admit that your opinion isn't the deciding factor in a situation.

The semantics of か, in both of these uses, is very much like the (literal) meaning of "probably" in English. You raise the thing as a possibility that you think is likely. This か probably historically derives from the usual question 〜か (or maybe 〜ではないか), but it feels pretty diverged at this point (i.e., そうする意味はないかと思います isn't equivalent to 「そうする意味はないか」と思います).