Lemon is twangier and tart. Lemons can make ones mouth pucker (even thinking about an extremely sour lemon can make some people scrunch up their faces and pucker). Fresh squeezed lemon juice is a great way to compliment certain foods and recipes. Lemon is delicious and I love fresh lemons.
Limes are similar but taste different. Limes are much sweeter and less sour. Although some things (such as Margaritas) taste sour, that's because of the other lemon-y/citrus-y ingredients, not the lime or lime juice.
But the above would be for actual lemons and limes. In a lip balm, the flavours may be quite different. I mean, I use cherry chapstick and it doesn't taste anything like a real cherry (but I still like it). I also have Dr. Pepper flavoured lip balm and it doesn't really taste like Dr. Pepper, either.
If I personally was deciding between lemon vs. lime-flavoured lip balm I would go for the lemon because it's likely more tart, more refreshing, and more tingly. They would both smell great (both lemons and limes smell fantastic)..so if you're basing this on smell (and not so much taste)..either one would be good.
ps
This is an incredible lemon lip balm. The taste, the smell, the balming effect and the packaging..
(highly recommend)
EOS lemon lip balm:
http://4.bp.blogspot.com/-TpqP61FN8zg/Tg48WwCVt4I/AAAAAAAAAc0/TXFE979krgo/s1600/rby-eos-lemon-drop-lip-balm-de.jpg