Many studies on mindfulness meditation, assessed in a review by Cahn and Polich in 2006, have linked lower frequency alpha waves, as well as theta waves, to meditation.[4] Much older studies report more specific findings, such as decreased alpha blocking and increased frontal lobe specific theta activity.[5] Alpha blocking is a phenomenon where the active brain, normally presenting beta wave activity, cannot as easily switch to alpha wave activity often involved in memory recall. These findings would suggest that in a meditative state a person is more relaxed but maintains a sharp awareness. Two large, recent comprehensive review works, however, point to poor control and statistical analyses in these early studies and comment that it can only be said with confidence that increased alpha and theta wave activity exists.[4][6]
Isochronic tones work just the same in delta as they do in alpha, theta and beta and they are widely used in the brainwave entrainment community to help people sleep. Like you, I’ve also seen some websites saying they don’t work in delta, but it’s a bit like the game of Chinese Whispers, where someone makes a comment and then after it gets passed around and shared a lot the message gets distorted and appears to be a fact. I don’t know of any scientific reason why they wouldn’t work in delta. I remember some people talking about this on a brainwave entrainment forum many years ago. They were saying they found isochronic tones a bit too abrupt for using to help them sleep and they preferred binaural beats, as they thought they were a more soothing sound. That was just a personal preference shared by a couple of prominent forum members at the time and some people then took that as a fact for everyone. That’s where I think that belief originated from.
But the question is: Does it have a special effect on the brain? A lot of people think so. The basic claim being made for binaural beats is "resonant entrainment". Entrainment, in physics, is when two systems which oscillate at different frequencies independently are brought together, they synchronize with one another, at whatever the combined system's resonant frequency is. Examples of entrainment occur in animals in nature; for example the chirping of crickets or the croaking of frogs. Synchronization of menstrual cycles in women is another example. Even people coming together and dancing with one another is a type of entrainment. The basic claim for binaural beats is that the perceived low-frequency beat will entrain your brain wave pattern, thus forcing your brain into some desired state.
In the meta-analysis performed by Fox et al., several sources of bias were indicated which bring into question the validity of meditation studies which use neuroimaging. Fox et al. suggests a publication bias may be leading to the over-reporting of significant results.[18] Despite this, however, Fox et al. found "consistent differences in prefrontal cortex and body awareness regions" in "areas key to meta-awareness..., exteroceptive and interoceptive body awareness..., memory consolidation and reconsolidation..., self and emotion regulation..., and intra- and interhemispheric communication..." and that changes were significant with "moderate" global median effect size and "consistent and medium-sized brain structure differences."[18]
So using the example track above, the right ear is sent a 20Hz beat, compared to a 10Hz beat in the left ear.  As the right ear receives the higher frequency of beat, this works to increase the speed of the ‘left' brain hemisphere, which can be helpful for people with conditions like ADD, who are often found to have an abundance of slow wave activity in the left brain.
A therapy that slows brainwave activity, helping to produce low frequency waves, is likely to aid relaxation and sleep. But it’s not only lowering brainwave frequency that binaural beats may offer to sleep and relaxation. A small study (19 people) has found that exposure to binaural beats is associated with changes to three hormones important to sleep and well being:
Effects on cortisol: The hormone cortisol is commonly associated with stress, and when levels are higher than usual, it can have a negative impact on the body. High levels of cortisol can lead to a range of conditions, including Cushing syndrome, menstrual cycle and female libido changes, anxiety, and depression. Dr. Giampapa noted a reduction of cortisol of up to 70-80 percent in several study participants.