That sentence indeed loses its point taken out of context. Condon and Cassen were not merely claiming that we can multiply a vector in $\mathbb{C}^2$ by an element of $\mathrm{SU}(2)$. They were claiming that when $\mathbb{C}^2$ is used as the Hilbert space for nucleons, this transformation is a symmetry of the laws of physics. In other words: if we have a solution of the equations of motion (whatever they are), we can apply this transformation and get a new solution.

More physically: since $\mathbb{C}^2$ is the Hilbert space for a nucleon, with one basis vector standing for a proton and one for a neutron, it means that we can do things like replace all the protons in the universe by neutrons and all the neutrons by protons, and this will have as little effect as, say, rotating the whole universe, or moving everything in the universe two feet to the left.

Of course this is false, taken literally. Protons and neutrons behave differently with respect to the electromagnetic force, since protons are electrically charged and neutrons are neutral! But of course Cassen and Condon knew this. They (and other people, like Heisenberg) were trying to focus on the nuclear forces and temporarily ignore the electromagnetic force. They hoped that in a universe like ours, but with the electromagnetic force turned off, and only the nuclear force remaining, the laws of physics would have this $\mathrm{SU}(2)$ symmetry.

If you read on, you'll see this hope turned out to be false. Nonetheless it was incredibly fruitful, because it got physicists thinking about Lie groups, and it got Yang and Mills to invent the Yang-Mills equations, which are very important in our current thinking about particle physics.

Summary: the difference is between a mere action of a group on a Hilbert space, and an action as symmetries of the laws of physics. That's why we said "symmetry group", not just "group".

Our sentence indeed loses its point taken out of context. Condon and Cassen were not merely claiming that we can multiply a vector in $\mathbb{C}^2$ by an element of $\mathrm{SU}(2)$. They were claiming that when $\mathbb{C}^2$ is used as the Hilbert space for nucleons, this transformation is a symmetry of the laws of physics. In other words: if we have a solution of the equations of motion (whatever they are), we can apply this transformation and get a new solution.

More physically: since $\mathbb{C}^2$ is the Hilbert space for a nucleon, with one basis vector standing for a proton and one for a neutron, it means that we can do things like replace all the protons in the universe by neutrons and all the neutrons by protons, and this will have as little effect as, say, rotating the whole universe, or moving everything in the universe two feet to the left.

Of course this is false, taken literally. Protons and neutrons behave differently with respect to the electromagnetic force, since protons are electrically charged and neutrons are neutral! But of course Cassen and Condon knew this. They (and other people, like Heisenberg) were trying to focus on the nuclear force and temporarily ignore the electromagnetic force. They hoped that in a universe like ours, but with the electromagnetic force turned off, and only the nuclear force remaining, the laws of physics would have this $\mathrm{SU}(2)$ symmetry.

If you read on, you'll see this hope turned out to be false. Nonetheless it was incredibly fruitful, because it got physicists thinking about Lie groups, and it got Yang and Mills to invent the Yang-Mills equations, which are very important in our current thinking about particle physics.

Summary: the difference is between a mere action of a group on a Hilbert space, and an action as symmetries of the laws of physics. That's why we said "symmetry group", not just "group".