Q: How do PUT, POST or PATCH request differ ultimately? The data, being sent over a PUT/PATCH/POST request, ultimately ends up in the database.
Now whether we are inserting a new resource or updating or modifying an existing one - it all depends upon the database operation being carried out.
Even if we send a POST and ultimately perform just an update in the database, it does not impact anywhere at all, isn't it?!
Hence, do they actually differ - apart from a purely conceptual point of view?

A: It is about the intent of the sender and from my perspective it has  a different behaviour on the server side.
in a nutshell:

*

*POST : creates new data entry on the server (especially with REST)

*PUT : updates full data entry on the server (REST) or it creates a new data entry (non REST). The difference to a POST request is that the client specifies the target location on the server.

*PATCH : the client requests a partial update (Id and partial data of entry are given). The difference to PUT is that the client sends not the full data back to the server this can save bandwidth.

In general you can use any HTTP request to store data (GET, HEAD, DELETE...) but it is common practice to use POST, PUT, and PATCH for specific and standardized scenarios. Because every developer can understand it later

A: 
Hence, do they actually differ - apart from a purely conceptual point of view?

The semantics differ - what the messages mean, and what general purpose components are allowed to assume is going on.
The meanings are just those described by the references listed in the HTTP method registry.  Today, that means that POST and PUT are described by HTTP semantics; PATCH is described by RFC 5789.
Loosely: PUT means that the request content is a proposed replacement for the current representation of some resource -- it's the method we would use to upload or replace a single web page if we were using the HTTP protocol to do that.
PATCH means that the request content is a patch document - which is to say a proposed edit to the current representation of some resource.  So instead of sending the entire HTML document with PUT, you might instead just send a fix to the spelling error in the title element.
POST is... well, POST is everything else.

POST serves many useful purposes in HTTP, including the general purpose of "this action isn't worth standardizing." -- Fielding 2009

The POST method has the fewest constraints on its semantics (which is why we can use it for anything), but the consequence is that the HTTP application itself has to be very conservative with it.
Webber 2011 includes a good discussion of the implementations of the fact that HTTP is an application protocol.


Now whether we are inserting a new resource or updating or modifying an existing one - it all depends upon the database operation being carried out.

The HTTP method tells us what the request means - it doesn't place any constraints on how your implementation works.
See Fielding, 2002:

HTTP does not attempt to require the results of a GET to be safe.  What it does is require that the semantics of the operation be safe, and therefore it is a fault of the implementation, not the interface or the user of that interface, if anything happens as a result that causes loss of property (money, BTW, is considered property for the sake of this definition).

The HTTP methods are part of the "transfer of documents over a network" domain - ie they are part of the facade that allows us to pretend that the bank/book store/cat video archive you are implementing is just another "web site".

A: They are slightly different and they bind to different concepts of REST API (which is based on HTTP)
Just imagine that you have some Booking entity. And yo perform the following actions with resources:
POST - creates a new resource. And it is not idempotent - if you sent the same request twice -> two bookings will be stored. The third time - will create the third one. You are updating your DB with every request.
PUT - updates the full representation of a resource. It means - it replaces the booking full object with a new one. And it is idempotent - you could send a request ten times result will be the same (if a resource wasn't changed between your calls)
PATCH - updates some part of the resource. For example, your booking entity has a date property -> you update only this property. For example, replace the existing date with new date which is sent at the request.

However, for either of the above - who is deciding whether it is going to be a new resource creation or updating/modifying an existing one, it's the database operation or something equivalent to that which takes care of persistence
You are mixing different things.
The persistence layer and UI layer are two different things.
The general pattern used at Java - Model View Controller.
REST API has its own concept. And the DB layer has its own purpose. Keep in mind that separating the work of your application into layers is exactly high cohesion - when code is narrow-focused and does one thing and does it well.
Mainly at the answer, I posted some concepts for REST.
The main decision about what the application should do - create the new entity or update is a developer. And this kind of decision is usually done through the service layer. There are many additional factors that could be done, like transactions support, performing filtering of the data from DB, pagination, etc.
Also, it depends on how the DB layer is implemented. If JPA with HIbernate is used or with JDBC template, custom queries execution...
