Why is the integral defined as the limit of the sum $\int_a^b f(x) dx = \lim_{n\to\infty}\sum_{i=1}^n f(x_i^*)\Delta x$? I am failing to understand why the integral is defined as:
$$\int_a^b f(x) dx = \lim_{n\to\infty}\sum_{i=1}^n f(x_i^*)\Delta x$$
instead of:
$$\int_a^b f(x)dx=\sum_{i=1}^\infty f(x_i^*)\Delta x$$

Is the former just popular preference or is there something I am not conceptually understanding here?

The nice thing about limits is that they can provide a rigorous answer in cases where you would otherwise get a nonsensical answer.
Consider the following function.
$$f(x)=\cases{5 &x $\neq$ 3\\ \text{undefined}&x = 3}$$
Which is the same as $$f(x)=\frac{5x-15}{x-3}$$
Based on the surrounding function a sensical value to fill the gap at $x=3$ would be 5. Limits are a way to find these sensical values (by sensical I mean they fit nicely with their surroundings). In this case it is pretty obvious:
$$\lim_{x\rightarrow3}f(x)=5$$
But the special thing about limits is that they don't actually need to know the value at the evaluation point $x=3$, the value that is given by the limit is solely determined by how the function behaves around that point. If you defined $f(3)=\pi^2$ or $f(3)=100$ instead the limit would remain unchanged.
So how does this connect to your question? Well your last sum doesn't actually make sense in itself:
$$\sum_{i=1}^\infty f(x_i^*)\Delta x$$
You are explicitly adding zero an infinite amount of times. What value is $x_3^*$? We don't know. The limit is defined though. Loosely speaking it is saying that when you increase $n$ the value of the sum gets closer and closer to a certain value. The integral is then defined* to be this value even though you never need to evaluate $n=\infty$. 
*The integral is ony defined if every possible limit gives the same answer, otherwise the integral is undefined.
