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.
As you say, a Wizard can learn spells from spellbook they find. So the question here is whether wizardly enemies will have spellbooks on them when they are defeated.
That's entirely up to the DM. Like any piece of loot, any enemy could be carrying a spellbook for whatever reason. If the DM wants the players to find the defeated Mage's spellbook, they will. There are a multitude of reasons for why a Mage would or wouldn't be carrying their spellbook on their person - it's the DM's choice to give the players access to this (fairly valuable) piece of treasure or not.
Remember that a Wizard doesn't require their spellbook to cast their spells (unless they want to cast them as rituals), so it's perfectly reasonable for a Mage to have hidden their spellbook somewhere once they prepared their day's spells.
So if you're asking this as a DM, the answer is that it's up to you. If you're asking as a player, the answer is that it's up to the DM, but it might well be worth your while to search the lairs/houses/dens/homes/castles/towers/whatever of enemy Mages you defeat.
Best Answer
Short answer: no; by the rules, you need to be a wizard to use a wizard’s spellbook.
There are two ways to build a character such as you describe.
The first is to have your character multi-class with wizard levels. To do this, you’ll need your DM’s buy-in to run a little adventure where you steal the spell book. Then you may have to work with your DM about what spells are in the book.
Backgrounds
Since you say “didn’t have enough money” and “stole a wizard’s spellbook” — in the past tense — I’m assuming none of this needs to happen during game play, but could be part of character creation.
In that case, consider simply choosing the Urchin or Criminal backgrounds, which will give you enough “rogue-like” abilities for your character concept to come alive.
Then, just start at wizard from level 1, with a good Dexterity stat, probably.
Backgrounds are really the system D&D 5e uses to enable character concepts like this.
Add a Bond or a Flaw
Put the story about the stolen spell book into the character background - it could be a bond (“I must redeem myself for stealing”), a flaw (“I know I don’t deserve to wear my wizard robes”) or both.
Starting Feat
If you can play a variant human, you can also pick a feat that jibes with your character concept.
The Urchin Wizard at my Table
In my first 5e campaign, I DM'ed for a player with a similar idea, who wanted to play a lost child who taught herself magic. The character took the Urchin background and the Wizard class.
It worked well evoking the feel you are going for: for the first couple levels, the stealth skills from the background were a significant fraction of the character’s abilities and “feel.” Eventually the wizard powers became “the main thing” but there were still those moments when another player would ask, “Wait, the wizard can pick locks?”