TL;DR: Spell scroll is a consumable item. It holds a spell which can be cast from it or copied into a wizard's spellbook, both of which destroy the scroll. Spell on a scroll refers to that specific spell, which is incidentally written on a scroll.
Spell scrolls (as you'd find in the treasure tables) are spells already prepared onto the scroll and contain some/all of the magic needed to cast them within them (which is why creating them is more than just copying things out of a book).
The second passage you are quoting describes how wizards copy any spell that they find written on a piece of parchment, in a book or on the back of a box of your favourite Orcish breakfast cereal, and put it into their spellbook.
Copying a Spell into the Book. When you find a wizard spell of 1st level or highter, you can add it to your spellbook if its of a level for which you have spell slots and if you can spare the time to decipher and copy it.
Copying a spell into your spellbook involves reproducing the basic form of the spell, then deciphering the unique system of notation used by the wizard who wrote it. You must practice the spell until you understand the sounds or gestures required, then transcribe it into your spellbook using your own notation.
The rules in the DMG are specifically about spells on spell scrolls (the type found in the random treasure tables etc.)
In addition to the rules on copying any spell into your spellbook you also have to follow these rules specific to spell scrolls.
A wizard spell on a spell scroll can be copied just as spells in a spellbook can be copied. When a spell is copied from a spell scroll, the copier must succeed on an Intelligence(Arcana) check with a DC equal to 10 + the spell's level. If the check succeeds, the spell is successfully copied. Whether the check succeeds or fails, the spell scroll is destroyed.
Basically you get one shot to understand enough about this spell scroll before the magic is used up and you can't copy it into your book.
So yes, there's a difference between a plain written spell on any old piece of parchment, and a spell scroll.
Additionally: As KorvinStarmast brought up in the comments you could have someone else help you with this check using the Help action.
Help
You can lend aid to another creature in the completion of a task. When you take the Help action, the creature you aid gains advantage on the next ability check it makes to perform the task you are helping with, provided that it makes the check before the start of your next turn.
Bare in mind that many DMs rule that you can only help with tasks that it makes sense for you to be able to aid someone with. In this instance I would think that at the very least they'd need to have the spell in question on their own class's spell list, or perhaps even be a wizard themselves.
Yes, spell scrolls are always destroyed by copying. From the DMG entry on spell scrolls, page 200:
When a spell is copied from a spell scroll, the copier must succeed on an Intelligence (Arcana) check with a DC equal to 10 + the spell’s level. If the check succeeds, the spell is successfully copied. Whether the check succeeds or fails, the spell scroll is destroyed.
(Emphasis mine.)
Best Answer
Yes!
The DMG states:
I tweeted the question to Mike Mearls, and he confirmed that as far as he knows spell scrolls are in fact consumed during the copying process. I will consider this to be the case until future published materials contradict Mike's assertion. Tweets provided below.