This is one vocabulary pair that has been giving me trouble for a long time. Both are pronounced かんしん. 感心 is admiration, 関心 is concern/interest.
And while I usually can infer a word’s meaning from the kanji it contains, unfortunately, “Feeling Heart” and “Connecting Heart” are not distinct enough for me to help me remember which is which.
I’d say 感心 means admiration, because when you admire someone your heart can beat so fast and hard that you can feel it. So you feel your heart.
For 関心, concerns and interests are things that are more about the mind. However, if you find someone with the same interests and concerns, you are likely to connect with them with your heart.
I usually think of 関心 as something that is “related to my heart”, thus is of concern to me.
The other word is 感心, which is sometimes used as interjection. For example, when you hear that someone has done something impressive, you could use it almost like you use “Bravo!” in English. “Oh, you summitted mount Everest last year? 感心、感心!”