People that want to convert you to their religion are usually concerned about YOU (saving your soul, etc.), so it's reasonable that it's YOUR choice to decline. The only concern is about your own well being.
People that want to convert you to veganism, on the other hand, are only concerned about the animals you're exploiting - it has nothing to do with you personally. Your choice to decline means you're condemning hundreds of animals to die every year for the rest of your life. This is a hard pill to swallow for animal sympathizers, so you must understand why arguments by vegans tend to be quite passionate.
But the two really aren't similar, other than the fact that they both make you uncomfortable.
The biggest hurdle from me going vegan was I thought it would be difficult, so I did something similar. Turns out, however, that it's SUPER EASY to cut out all meat and dairy. Seriously it blew my mind. Just look up a couple recipes, make sure you read ingredient lists on products, and you're done. That's all it takes.
The key is to just commit. Jump off the high board and take the plunge. Sure the water may shock you at first but you'll quickly adapt. Quicker than you'd expect.