Is there a difference between "这里 的 天气跟 我 老家 一样", and "这里 的 天气跟 我 老家 的 一样"
Well actually not so big difference as we could use either of them in daily conversation. Maybe in your writing it would be better to use the latter but we usually save characters in chat as long as it's understandable by others.
Ps: You can even omit the first "的" to make the sentence shorter.