Really? Olga was the eldest, and the most independent, but I think Tatiana had a more mature attitude towards things and didn't let things weigh her down, like they could do to Olga. Tatiana had a strong sense of duty, much like her loved mother. That's why Tatiana gives me the impression of being like the eldest, or the "leader". I think that sort of character is very admirable, and I admire Tatiana even more because she was also very devoted and affectionate.
As for Olga, I think she probably deeply appreciated her sister for being able to take charge of things - not to say she was lazy, but it may not have been her "thing" or greatest talent to take responsibility of households and know what practical things to do immediately in emergencies. But like Holly said, it's not like one of the girls was better than the other. Both of them had very admirable qualities and I admire both sisters very much.