Aside from multi-classing and taking the Magic Initiate feat I know you are generally limited to the class list. But what about scrolls? It says in the PHB that you can learn spells that are found during your adventures if you spend the time and components (50 gp per level of spell) to learn the spell and copy it into your spellbook. In LMOP you can find a scroll of revivify. If the spell can be written down on a scroll, why wouldn't a wizard be able to copy that into his spellbook?
From the "Your Spellbook" sidebar, page 32 of the Player's Basic Rules (v0.2):
When you find a wizard spell of 1st level or higher, you can add it to your spellbook if it is of a level for which you have spell slots and if you can spare the time to decipher and copy it.
Emphasis mine. Only spells on the Wizard list can be learned in this way. In this area, there is no difference between spells written in a spellbook and spells written on scrolls. The difference is between Wizard spells and non-Wizard spells.
Even if the game allowed you to copy a non-Wizard spell into your spellbook, we have in Preparing and Casting Spells, page 30:
You prepare the list of wizard spells that are available for you to cast. To do so, choose a number of wizard spells from your spellbook equal to your Intelligence modifier + your wizard level (minimum of one spell).
So even if you could write non-Wizard spells in your spellbook, you couldn't prepare them and therefore couldn't cast them.
Further down the same page, you can cast rituals without preparing them, but even there, we have:
You can cast a wizard spell as a ritual if that spell has the ritual tag and you have the spell in your spellbook.
So, you can't copy non-Wizard spells into your spellbook, and even if you could, there's just no way to cast them.
Wizards can't use scrolls that are not on their list either, since we have on page 60 of the DM's Basic Rules:
If the spell is on your class's spell list, you can use an action to read the scroll and cast its spell without having to provide any of the spell's components. Otherwise, the scroll is unintelligible.
As to why this is (from a narrative perspective), every method of spellcasting works differently. For example, the Wizard and the Sorcerer probably have the most similar spell lists, but the Wizard uses magic through painstaking study and preparation, where a sorcerer simply unleashes the power they have within them. So a spell that isn't on the Wizard spell list is probably just impossible to cast using the Wizard method of spellcasting.
From a balance perspective, obviously it would be completely unfair if Wizards had access to every spell in the game.