For example, what is the difference between these sentences?:
Don't they both mean "There are many shrines in Japan"?
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I copied these sentences from the other answer, who was correct to point out the grammar errors. There is, however a difference in meaning between these two sentences. In written, concise English, you may get the same translation for these sentences, but they are not the same thing in Japanese. To put is simply, it's a matter of what is being emphasized. I'll keep my answer focused on the difference between は and には. If you have other questions about grammar, please feel free to ask in the comments to this answer.
は functions as the Japanese topic marker. This is distinct from the subject, in that it is a little less for specific. You'll encounter the peculiarities of this when studying the difference between は and が, but I'll leave that to others to explain it to you. To keep this short, you can treat は as saying, "as for ___."
So when we deconstruct the first sentence, 日本は神社が多い, we get:
Japan (topic) shrines (subject) are numerous. The verb appears left off, as it is actually part of the adjective 多い, but we can safely say that it is 'to be.' To put it in more natural English that closely mirrors the Japanese, we would get:
As for Japan, there are numerous shrines.
Because of the use of は, we are not making a comparison to any other countries or regions. We are only commenting that there are many shines in Japan.
には is actually a combination of two particles. In this case, に indicates a place of existence (there are many other meanings for this particle), and は functions similarly to what we previously discussed. In many cases, に can be translated to 'in,' but it is not a 1 to 1 translation.
So what's the topic of this sentence? Believe it or not, it's actually 日本に. So breaking down the second sentence '日本には多くの神社があります', we get:
In Japan (topic), numerous shrines (subject) exist. In more natural English, we get:
In Japan, there exist numerous shrines.
Because these sentences are so similar in translation, frequently the more concise translation:
There are many shrines in Japan. It's not uncommon for subtleties of the Japanese language to be lost in translation.
Consider what would change in the meanings if we left out 日本.
Originally, we have 日本は神社が多い。 日本には多くの神社があります。
After we take out 日本、
Shrines (are) numerous.
(There) exist numerous shrines.
Because が is used to identify something, the subtle difference is that in the first sentence, we're identifying shrines as the things that are numerous (in Japan). The second sentence identifies something existing in Japan. Numerous shrines exist in Japan.
It may be easier to understand the subtlety by leading these sentences with questions. "What is numerous in Japan?" and "What's in Japan?" would work well to lead these, respectively. However, after leading with those questions, Japan is already known as the topic, so the sentences I gave, without mentioning Japan, would likely be used so as not to repeat the context.