These are equivalent primitives assuming the existence of one-way functions, which implies $\mathbf{P}\neq\mathbf{NP}$$^*$. It was shown in [G+,SW] that IO plus OWFs implies public-key FE.$^{**}$ The converse, that sub-exponentially-secure public-key FE (with some succinctness property) implies IO, was shown in [BV].
On the other hand, as pointed out in the comment by @integrator, if $\mathbf{P}=\mathbf{NP}$ then IO exists (simply pick the smallest/lexicographically-first circuit which computes the same function) but FE (which implies PKE) does not.
$^*$This was relaxed to $\mathbf{NP}\not\subseteq \mathbf{io}- \mathbf{BPP}$ in [K+].
$^{**}$[G+] assume PKE and NIZK in addition to IO. These were later shown to be implied by IO and OWFs [SW].
[BV] Bitansky and Vaikuntanathan, Indistinguishability Obfuscation from Functional Encryption, FOCS'15
[G+] Garg et al, Candidate indistinguishability obfuscation and functional encryption for all circuits, FOCS'13.
[K+] Komargodski et al, One-Way Functions and (Im)perfect Obfuscation, FOCS'14
[SW] Sahai and Waters, How to Use Indistinguishability Obfuscation: Deniable Encryption, and More, STOC'14
