Phosphorescence and delayed fluorescence are not the same thing. Both phosphorescence and fluorescence are radiative deactivations of electronically excited states in organic molecules, processes that can be lumped together in the more general category of luminescence. The primary distinction is the quantum electronic spin character of the emitting excited state. Formally, fluorescence is a process in which the electronic excited and ground states are the same and phosphorescence is a process in which the states have different spins. The overwhelming majority of organic luminophores have singlet ground states, and so the key excited states of fluorescing molecules are also almost always singlets. In most cases of phosphorescence, the emitting excited state is a low-lying triplet. Because a transition between a singlet and a triplet (intersystem crossing) is formally forbidden, the radiative rates involved in phosphorescence are orders of magnitude slower (typically a millisecond or more) than those in fluorescence (nanoseconds or lower). Note that the phosphorescence kinetics are limited not only by the rate of intersystem crossing from the excited triplet to the ground singlet, but also the process of intersystem crossing from the excited singlet to the excited triplet. The phenomenological distinction between phosphorescence and fluorescence is that the latter, because it is such a slower process, often manifests itself as delayed. When the phosphorescence lifetimes are long enough, such as in zinc sulfide "glow in the dark stickers", the material appears to emit light long after the excitation source is turned off. In point of fact, every photoluminescent process, be it phosphorescence or fluorescence, involves a delay. It is just the case that in fluorescence, the delay is so short that you cannot observe it without sufficiently fast-responsive equipment, which doesn't include your eyes. Most phosphorescent materials are also too fast to observe this delay, being on the order of milliseconds. Nevertheless, one of the most definitive ways to distinguish between phosphorescence and fluorescence is to measure the kinetics of the process, although usually an assignment can be made also based on the energy difference between the absorption and emission bands (phosphorescence typically involves a much larger Stokes shift, because of the large energy difference between the singlet and triplet states).
A few other things are noteworthy here.
1. Delayed fluorescence is a process in which the excited triplet state can actually convert back to an excited singlet state with a rate that is competitive with phosphorescence. The re-prepared singlet state can then itself fluoresce. So while the timescales for delayed fluorescence are significantly longer than a direct fluorescence, and may be confused with phosphorescence based on timescales alone, it is in fact a singlet-singlet transition and therefore formally fluorescence. Delayed fluorescence is easily distinguished from other competitive photophysical process by time resolved fluorimetry. And in any case it is pretty uncommon to observe it.
2. The observation of delayed emission may also be chemical in origin, where the limiting factor is reaction kinetics. The luminol reaction is a good example. Here the process is formally fluorescence, as the emission occurs from an excited singlet state that is prepared chemically. The observed light emission is delayed, however - not by the physical nature of the singlet state, but the rate at which the excited singlet state is chemically produced.
3. As mentioned in my earlier post, the terms are largely phenomenological and due to the historical development of photochemistry/physics, they are also confusing. The modern formal definitions of the term are still broadly useful but are often misapplied, even by experts, particularly in relation to materials that are not easily classified as organic chromaphores, around which the field of photochemistry and photophysics was historically developed. In particular, fluorescence and phosphorescence are often applied in situations where the emitting state is not easily classified as formally a singlet state or triplet state, respectively. A good example of the latter is the intensely luminescent Ru(bpy)32+, and related metal polypyridyl complexes, which have low lying excited states characterized by transfer of electron density from the metal center to the polypyridyl ligands. These states are often formally assigned whole-number spin designations. The lowest lying state is often labeled as a triplet state, and indeed it behaves as such under many relevant tests (quenching by oxygen, e.g.). But the photoluminescent lifetime is only on the order of microseconds, much shorter than those characteristic of phosphorescence from pure triplets of organic molecules. Moreover as the ligand structure is altered, the excited state may become more like a pure charge transfer state. Luminescence properties attenuate accordingly. At what point can we no longer really call this a triplet state? In my opinion, it's more appropriate to label emission from these complexes as luminescence rather than phosphorescence, and yet the latter term persists. Maybe an even more salient example is in the treatment of quantum dot luminescence. Here to my mind it is even less clear that the emitting state is a singlet state in the traditional organic framework sense. Part of the trouble here is the continued lack of harmonization between inorganic semiconductor (solid state physics) and organic material (physical chemistry) language sets. Semiconductor terminology such as excitons and confinement can of course be applied to organic molecules, but this is still not very straightforward or standardized and I see the same issues with applying organic photophysics terminology (like fluorescence and phosphorescence) to inorganic semiconductors. Does a quantum dot emit by fluorescence? Much of the literature would have you think so, but I see using this term as being at best misleading, and for my part I try to use the word "luminescence" instead, which is more general. (For full disclosure, I have been away from the field for a few years and haven't kept up with the latest understanding of the physics of these materials, so maybe this situation has changed.)
I guess the point is that I would be careful about these terms when they are used in any context not related to a conventional organic chromophore, and the timescale for luminescence is not by itself a perfect indicator of the underlying physics and chemistry. Nevertheles, in the case of organic molecules phosphorescence and fluorescence have pretty well understood physical origins and can be applied with confidence to indicate the nature of the emitting states and the optical properties such molecules are likely to manifest.