There is a clear, self-contained classification of compact, connected Lie groups in "Lie Groups: An Approach through Invariants and Representations" by Claudio Procesi. See Chapter 10, Section 7.2, Theorem 4, page 380. It says such a group is of the form $K_1\times\dots\times K_n\times T/Z$ where each $K_i$ is connected, compact, and simply connected, $T=(S^1)^m$ is a compact torus, and $Z$ is a finite subgroup of the center satisfying $Z\cap T=1$. Furthermore two such groups $K_1\times\dots\times T/Z$ and $K'_1\times\dots\times T'/Z'$ are isomorphic if and only if there is an isomorphism $K_1\times\dots\times T\rightarrow K'_1\times\dots\times T'$ taking $Z$ to $Z'$. Note that the numerator is specified by a list of types $A_n,B_n,\dots, E_8$ and the integer $m$. The result is obtained from a bijection between compact connected and reductive algebraic groups, and is equivalent to the classification by root data.

For example there are precisely $3$ compact groups of rank 2 and semisimple rank 1: 

$SU(2)\times S^1$ ($Z=1$)

$SO(3)\times S^1$ ($Z=\langle -I,1\rangle$)

$U(2)=SU(2)\times S^1/\langle (-I,-1)\rangle$.

For another example the following two groups are not isomorphic, where
$\zeta=e^{2\pi i/5}$:

$SU(5)\times S^1/\langle(\zeta I, \zeta)\rangle \simeq U(5)$

$SU(5)\times S^1/\langle(\zeta I,\zeta^2)\rangle$