In your case, it sounds like its better to agree to disagree, but that shouldn’t stop you being best friends.
In answer to your more general question, I’ll tell you of a situation I was in recently. I had a ‘friend’ who I hung around a lot, mainly because I could tell he had trouble making friends and I felt sorry for him. We hung out for about four months as quite good friends before things started getting strange. I met a girl I was interested in, and decided that I would hand out with that group to see if there was any potential. He began to feel threatened though, and tried to avoid the group and drag me away.
When I started dating this girl, he started complaining that I wasn’t paying enough attention to him, and I wasn’t talking to him every time I was online. I tried to gently tell him to back off a little, and give me space. This distressed my girlfriend, who thought she was destroying my friendship.
After a while trying to balance things this way, he started buying me presents. More than I have bought for my girlfriend to this day. I decided to get rid of him, but I was unsuccessful because I underestimated his possessive nature. The final straw came when he bought a card, supposedly for me to give to my girlfriend for Valentines Day (which I promptly disposed of without looking at). From that point onwards, I told him firmly to never contact me again, ignored his communications, hung up on him when he called, and told him to get off my property when he came to offer more presents.
The upshot is, if you are uncomfortable hanging around someone, don’t. If there is a valuable friendship there to save, save it. The action is easy, the decision to take action is hard. Just ask yourself, will this be an issue in six months time? Do I still want to be friends? What am I going to do about it?