Connections between finite and Lie groups in Fulton and Harris In Representation Theory: A First Course by William Fulton and Joe Harris, the following appears on the first page about Lie Groups

From a naive point of view, Lie groups seem to stand at the opposite end of the spectrum of groups from finite $\text{ones}^1$

with the superscript referencing the following note

$^1$ In spite of this, there are deep, if only partially understood, relations between finite and Lie groups, extending even to their simple group classifications.

Would someone shed a little more light on this? Specifically:


*

*Why is it naive to think of Lie Groups as being at the opposite end of the spectrum of groups from finite groups? 

*What are the relations between finite and Lie groups that the authors are referencing?  

*In what way are these relations only partially understood? 
 A: Here are comments on your three questions:
(1) "From a naive point of view" I would interpret this comment in the topological sense, finite sets are topologically very different to Lie groups which are differentiable manifolds.
(2) Well for one thing, the study of representation theory (over $\mathbb{C}$) of finite groups is a question of finite subgroups of $\operatorname{GL}_n(\mathbb{C})$. 
Then what I would personally care more about: besides the alternating groups, almost all finite simple groups are finite groups of Lie type. These arise as groups which are analogous to Lie groups, e.g. $\operatorname{SL}_n(\mathbb{C})$ and $\operatorname{SL}_n(\mathbb{R})$ are analogous to $\operatorname{SL}_n(\mathbb{F}_q)$ (where $\mathbb{F}_q$ is a finite field of order $q$).
The structure of these finite Lie groups is studied in very much the same way: we have root systems, maximal tori, Borel subgroups, parabolic subgroups, etc. The representation theory is more complicated but is still analogous in some sense. For example, the finite-dimensional irreducible representations $\rho: \operatorname{SL}_n(\mathbb{C}) \rightarrow \operatorname{GL}_{d}(\mathbb{C})$ are determined by their "highest weights", and similarly for for finite-dimensional absolutely irreducible representations $\rho: \operatorname{SL}_n(\mathbb{F}_q) \rightarrow \operatorname{GL}_{d}(\mathbb{F}_q)$.
Also, the results established for Lie algebras and Lie groups over $\mathbb{C}$ often help us with results for finite groups of Lie type (sometimes even conversely, I think).
(3) Too vague to say much, but it is certainly true that especially for finite groups of Lie type, their representation theory is closely related to that of Lie groups. Even today, there are still many open questions here.
