How is IOTA's Tangle different than classical blockchains? How is IOTA's Tangle different than classical blockchains? What are the main differences and how do they translate to real world usage? What are the strengths and weaknesses of the Tangle?
Bitcoin is based on a Blockchain approach, meaning we build a chain of blocks, each referencing the previous block. For the creation of each block, a Proof of Work (PoW) needs to be done by miners. These blocks are a means to reach consensus which transactions have taken place and can be considered safe (e.g. 6 blocks)

IOTA is based on a Distributed Acyclical Graph (DAG) approach, which they call the "Tangle". No more blocks, no more mining, instead when you broadcast a transaction, you first need to do PoW validating two previous transactions. You "attach" your transaction to those two previous transactions (see image)



The consensus about a valid transaction is not reached by number of blocks, but applying the socalled Random Walk Monte Carlo Algorithm. More details can be found here: https://forum.iota.org/t/iota-consensus-masterclass/1193 
In short: your transaction is considered valid (green in the image) if you can reach it from every unconfirmed transaction (grey).

Since there is no necessity for blocks, there is no issue with the block size (see scaling debate). The theory - which remains to be practically proven - is that a DAG can hypothetically scale infinitely and that a greated number of transactions actually makes the network faster, since a transaction means confirmations.

For more info, I recommend the following videos:
https://www.youtube.com/watch?v=UwEp5cexTJE&amp;t=251s
https://www.youtube.com/watch?v=MsaPA3U4ung&amp;t=3s