First employee and architect of a major cryptocurrency exchange here. (Disclaimer: I am not a mathematician.)

IMHO the term "blockchain", in current and common use, actually means Merkle trees applied to time-series information that is shared between multiple computers over the internet, hence 'distributed database'.

Merkle trees are basically a way of saying "this precise set of information was used to generate this new precise set of information". Applied to successive timestamped data in a blockchain, this means that participants within a network can provide strong evidence for past state and state change - ie. they establish relative trust in the fundamental integrity of a shared and evolving database (the blockchain) based upon mutually agreed rules.

What is the real world advantage? Frankly, in most cases, there isn't one, but there are definitely disadvantages. Distributed databases are not new, and any other way of executing them is demonstrably more efficient, simple, etc.

The main potential advantage is to remove the ability of a central authority (or small colluding group of bad actors) to "change the rules" by "rolling back" the state of the system, halting the system, seizing control, removing actors, etc.

This property is perhaps at its most potentially impressive when its implications with respect to the conventional business environment, government and politics are considered for large global industries with deep pockets. Simply speaking, right now most regulations are issued in airy-fairy lawyer speak and cost a lot of money to skew/change/interpret/argue/take to court. This is the case for government regulation, case law and non-governmental agreements (industry codes of conduct, informal protocols, MOUs, contracts, SLAs, etc.).

The potential advantage of distributed systems with mutual trust (note that blockchains are only one implementation of this concept) are that all of this overhead can be removed, since an adequately cross-border (ie. cross-jurisdiction, or government-independent) network of an adequate size will be essentially immune from direct state actor intervention, and the rules for all parties remain clear.

To summarize: save lawyer money, ignore regulators, save time/hassle, remove any means of effective government intervention.

The reality, however, is that most industries are not able to agree on a clear set of interactions and rules under which to formalize their activities, and even if they could it would probably not be feasible to do so in all cases, therefore this semi-utopian objective cannot be reached, and in my professional opinion few corporate blockchain projects to date are delivering anything more than an inefficient means of distributing simple time series transaction data.

The big exception (ie. winning example for your last question) is Bitcoin, IMHO because it was the first, and because it quickly enabled safer and more cost effective means of recreational drug distribution, which essentially provided the initial market which funded its growth (now tending to speculation).

My prediction is that we will see a rise in smaller, private systems and asset types providing transaction services with different properties to present era cryptocurrencies and a smart layer will emerge to evaluate and utilize these based upon their objective properties and per-transaction requirements / risk models. Many of these systems will not be based on blockchain at all. Instead of de-facto bank or card systems, we will have a broad range of settlement mechanisms available. Instead of stock exchanges or large VCs, we will have a broad range of funding sources available. Access to capital will improve, business transparency will improve, government regulators and established financial services monopolies will slowly be disempowered.

Further reading - https://en.wikipedia.org/wiki/Merkle_tree + https://en.wikipedia.org/wiki/Blockchain + https://en.wikipedia.org/wiki/Time_series + https://en.wikipedia.org/wiki/Distributed_database + https://en.wikipedia.org/wiki/Bitcoin_network + https://en.wikipedia.org/wiki/Money + https://en.wikipedia.org/wiki/Cryptographic_hash_function