From a practical standpoint...
LDA starts with a bag-of-words input which considers what words co-occur in documents, but does not pay attention to the immediate context of words. This means the words can appear anywhere in the document and in any order, which strips out a certain level of information. By contrast word2vec is all about the context in which a word is used -- though perhaps not exact order.
LDA's "topics" are a mathematical construct and you shouldn't confuse them with actual human topics. You can end up with topics that have no human interpretation -- they're more like artifacts of the process than actual topics -- and you can end up with topics at different levels of abstraction, including topics that basically cover the same human topic. It's a bit like reading tea leaves.
I've found LDA useful to explore data, but not so useful for providing a solution, but your mileage may vary.
Word2vec doesn't create topics directly at all. It projects words into a high-dimensional space based on similar usage, so it can have its own surprises in terms of words that you think of as distinct -- or even opposite -- may be near each other in space.
You can use either to determine if words are "similar". With LDA: do the words have similar weights in the same topics. With word2vec: are they close (by some measure) in the embedding space.
You can use either to determine if documents are similar. With LDA, you would look for a similar mixture of topics, and with word2vec you would do something like adding up the vectors of the words of the document. ("Document" could be a sentence, paragraph, page, or an entire document.) Doc2vec is a modified version of word2vec that allows the direct comparison of documents.
While LDA throws away some contextual information with its bag-of-words approach, it does have topics (or "topics"), which word2vec doesn't have. So it's straightforward to use doc2vec to say, "Show me documents that are similar to this one", while with LDA it's straightforward to say, "Show me documents where topic A is prominent." (Again, knowing that "topic A" emerges from a mathematical process on your documents and you then figure out what human topic(s) it mostly corresponds to.)