Translation:I do not like spring but, on the other hand, I like summer.
You use both words in two different cases, and they pretty much have the same meaning.
мне нравиться-- This is literary "at me, it is pleasing"
я люблю-- This is literally "I like/love"
In my experience (Keep in mind, I am still a beginner) I heard "мне нравиться" way more than "я люблю" when referring to "I like something." And most books and podcast that I use uses the мне нравиться as the go to example for "I like." For some reason, Duo is using "я люблю" to introduce us to the "I like something." Maybe because it is easier to explain.
Personally, I use "мне нравиться" for "I like" and "я люблю" for "I love." Simply because it sticks to my brain better that way, and it helps me to figure out more complicated Russian sentences like "Мне больше нравится" (At me, it is more pleasing/I prefer) Мне больше нравится кофе без молока/I prefer coffee without milk.
A russian friend of mine told me to try to use нравится whenever I referred to objects (and she did so way more bold than I'd have expected), in order to avoid sounding like I am in love to an object (люблю). She used "i love that picture" as an example, and told me to never say "я люблю этот фото" but "мне нравится этот фото" (I am a beginner, please excuse any mispellings)
Either sounds OK to my ears (native UK speaker). I think the article is optional. Thinking about recent changes in the weather, I might say: "I don't like autumn", but I don't think I would sound strange or crazy if I said: "I don't like THE autumn", either. Similarly, "The leaves change colour in autumn", OR: "The leaves change colour in THE autumn." Again, either OK.
Just to be clear, I love the Russian language. The more I learn the more I see how subtle it is. But I love English too and the translation above is what I would call "tortured" - no one would ever say it like that. I'm guessing the use of зато rather than но conveys a stronger sense of "BUT," a stronger contrast. In English we accomplish that by the stronger "love" which we use for inanimate things to mean "really, really like." So, "I don't like spring but I love summer" is what I think the person is saying.
I think "however" should work here, but "although" is a closer translation of "зато." "However" implies a (possible or slight) contradiction of the preceding, whereas "although" implies that "and it follows that" the preceding. Even the word "зато," broken down to "за то" means "after that" or "it follows that."