Connective spectra and infinite loop spaces

Ok, this discussion has grown beyond the level of comments so I'll collect the facts here. A bit of terminology: a $(-1)$-connected space is a space with a choice of basepoint and the category of $(-1)$-connected spaces $Top_{>-1}$ is the category of pointed homotopy types with basepoint-preserving maps. The category of $n$-connected spaces $Top_{>n}$ is the full subcategory of $Top_{>-1}$ of spaces $X$ which have all homotopy groups $\pi_k(X) = 0$ for $k \leqslant n$. For any space $X$ we can take its $(n-1)$-truncation $X_{<n}$ which has all homotopy groups $\pi_k = 0$ for $k\geqslant n$. There is a fiber sequence $X_{\geqslant n} \to X \to X_{<n}$ with the right arrow an isomorphism for $\pi_{k<n}$ and the left arrow an isomorphism for $\pi_{k\geqslant n}$.

Now, informally a loop space $X$ is just a space with a homotopy equivalence $X \simeq \Omega Y$ for some other space $Y$, however this definition is not precise enough since there can be many non-equivalent choices of $Y$. We would want some structure on $X$ that would guarantee it is a loop space without explicitly specifying such $Y$. A famous theorem of May tells us that that a space is a loop space if an only if it is grouplike and is equipped with the structure of an algebra over the $A_\infty$ (aka $E_1$) operad, in particular it is $(-1)$-connected. In this case there is a canonical choice of delooping denoted $\mathbb B X$ which is a 0-connected space such that $\Omega \mathbb B X \simeq X$ as homotopy types and as $A_\infty$-spaces. Even more specifically, the functors $\Omega$ and $\mathbb B$ are well-defined on the categories of $(-1)$-connected spaces and $A_\infty$-spaces respectively and establish an adjunction $\mathbb B \dashv \Omega$ between them, which restricts to an equivalence between the full subcategories of grouplike $A_\infty$-spaces and $0$-connected spaces. In particular, $\mathbb B X$ is the unique 0-connected delooping of $X$, and for any other delooping $Y$ its 0-connected cover $Y_{>0}$ is thus equivalent to $\mathbb B X$. Thus we don't need to consider non-connected deloopings when defining loop spaces.

So formally we can define a loop space as a grouplike $A_\infty$-space, and a map of loop spaces as the map of spaces preserving the $A_\infty$-structure (grouplikeness is a property rather than structure so doesn't impose extra conditions). These are exactly the maps induced on the loop spaces from the maps $Y \to Y^\prime$. Note that there can be many different $A_\infty$ structures on the same topoolgical space, thus many nonequivalent deloopings (a classic example is $S^3$ which have different $A_\infty$ structures different from $SU(2)$ group structure arising from mixing various structures at primes). For this reason it doesn't make sense to just say that a loop space is $X$ such that there exists $Y$ with $\Omega Y \simeq X$ --- the specific choice matters.

Generalizing to all $n$, an $n$-fold loop space can be delooped $n$ times and the category of $n$-fold loop spaces is equivalent to the category of $(n-1)$-connected homotopy types. An $\infty$ loop space is thus a space $X$ such that it can be delooped an arbitrary number of times. We can either define it as an algebra over the $E_\infty$ operad (which is a colimit of all $E_n$ with $E_{n+1} = E_n \otimes E_1$) or as a sequence of spaces such that $n$-th space is $(n-1)$-connected and $Y_n = \Omega Y_{n+1}$. Such sequences are exactly connective spectra.

The definition via an arbitrary sequence $Y_i$ requries a different notion of equivalence, since such sequences represent arbitrary spectra. We can say that a map of two spectra $Y_i$ and $Y_i^\prime$ is an equivalence if it is a homotopy equivalence on the 0-th component. Note that the sequence of $(i-1)$-connected covers $(Y_i)_{>i-1}$ is also a spectrum and its canonical map to $Y_i$ is an equivalence of $\infty$-loop spaces. This construction realizes the subcategory of connective spectra as the localization of the category of all spectra. I understand this is Adams's approach in Infinity loop spaces.

A simple way to answer this question is: denote by $Spec$ the homotopy category of spectra, $Spec _0$ the category of (-1)-connected spectra, $InfL$ be the homotopy category of infinite loop spaces. The assertion that "connective spectra are the same as infinite loop space" simply means that the composition of functors $$Spec _0 \stackrel{i}{\to} Spec\stackrel{\Omega ^{\infty}}{\to} InfL$$ is an equivalence, where $i$ denotes the inclusion. You gave an example of infinite loop space in the image of $\Omega ^{\infty}$, which doesn't mean that it is not in the image of $\Omega ^{\infty}i$.