Why can there be an infinite difference between two functions as x grows large, but a ratio of 1? I learned in grade school that the closer $a$ and $b$ are to one another, the closer $\frac{a}{b}$ is going to be to $1$. For example, $\frac{3}{\pi}$ is pretty close to 1, and $\frac{10^{100}}{42}$ isn't even close to 1.

So, why is:

$$\lim_{x\to\infty} \frac{x^{2}}{x^{2}+x} = 1$$

But:

$$\lim_{x\to\infty}[(x^2+x)-(x^2)] = \infty$$ ?

Seems pretty counterintuitive. What's going on here?
Vadim's answer handles the math (and I've upvoted it), so I will try to provide intuition.

The idea is the word "closer" is relative.  That is, in some sense, $100{,}000$ is closer to $100{,}010$ than $1$ is to $0$.  Of course, in an absolute sense, $1$ is $1$ away from $0$ while $100{,}000$ is $10$ away from $100{,}010$.

But, say you had $\$1$ and I took away $\$1$ and contrast that with the scenario where you have $\$100{,}010$ and I take $\$10$.  In the second scenario, I take more money.  But in the first scenario, you care a lot more about the theft.

In your particular example, something similar happens.  Yes, $x^2 + x$ and $x^2$ have a very large difference when $x$ is large.  But the relative difference is tiny.  For example, when $x = 1{,}000$, then $x^2 + x = 1{,}001{,}000$ while $x^2 = 1{,}000{,}000$.  So, yes, the difference is large (it's $1000$), but the relative difference is not that big - these two numbers are separated by a mere $0.1\%$  And this percentage shrinks as $x$ gets even bigger.
Trying to develop an intuitive idea using images and, doing so, cutting a lot of corners. 

Suppose Bill Gates gets a raise of 1 dollar an hour. And suppose he decides to work forever. He will make 1$ more every hour. So an infinite amount of money more than without the raise. But do you think this raise is significant? No. The amount of money he makes with or without the raise is almost the  same. The ratio between the fortunes he amasses is pretty close to one.