I have been reading quite a few varied opinions about this and given it some thought,  I will set out what seems to me logical and let you make your own decisions!

First we must allow all other things to be equal, otherwise confusion will reign. We will use a notional 500mm lens with a barrel mounted tripod mount. we will set it at infinity and F8. We will use two camera bodies, one APS, one full frame, the density of the megapixels will be the same, lets say 10 megapixels for the APS and 15 for the full frame, which I think gives about the same density. (If you have one camera with denser "pixels" than the other, you have an advantage specific to that individual sensor/camera, not to the format).

Basics
You are no doubt already aware that to compute the comparative field of view of a FF lens used on an APS camera body you multiply by 1.5 (usually, it may be from 1.3 to 2.0). This probably happened because FF sensors were very expensive - it would seem logical to me that given the choice, camera makers would have gone for equivelence if it had been practical, to ease the transition and gain customer acceptance.

Does the APS sensor give more telephoto reach?

No, how could it? The lens is the same. What has happened when you swop over the bodies without touching the lens is you only used part of the image the lens projected into the camera body, you cropped it. If the sensors are of the same density you will get exactly the same effect by post shoot cropping in Photoshop, The extra magnification is no more real than in a "digital zoom" on a compact camera.

On the other hand, as you work up through the formats towards meduim and large format you need larger and larger focal lengths to achieve the same field of view. Nobody talks about cropping and applying adjustment factors in those formats, thats partly because the lenses are always designed to match the sensor size. The option of using a cropped version of the potential image is probably unique to the current FF/APS SLR situation. 
However, ask yourself this, if somebody said I need a shot of that faraway object, would you suggest a smaller sensor or a longer lens as the best answer? 
I think the way to look at it is to accept that the 500mm lens from 35mm/FF days, mounted on an APS body,  has the field of view you would have expected of a 750mm, but not to be fooled that you suddenly gained any quality or reach.

Is there more depth of field with an APS sensor?

Well, see above, how could there be, its the same lens? But wait a minute, we must consider real world situations. In the real world, you look through a lens, you see the smaller subject matter image with FF, what do you do? You zoom in or you move closer, maybe you add a teleconverter. If you do any of these things with the FF body you will then have shallower depth of field. I will let you decide if you think that justifies saying APS has more depth of field in that context.
In general, APS lenses have shorter focal lengths for a given field of view, so they will have more depth of field in any given situation, much as medium format cameras have less. Basically, as you grow the captured image size you lose depth of field.  

What are the true advantages of APS over FF?

Size, weight and cost. Especially if you only shoot APS and have "digital only" lenses which produce images of the correct size for APS. Much of the confusion around FF and APS is caused by the fact that unlike other formats, lenses are available that work (or nearly work) with both sizes. It is important to remember that when using an FF capable lens with an APS body you effectively have an oversized lens that is making an unnecessarily large image in the camera. The Sigma 12-24 FF weighs about 600g and the 10-20 APS only weighs 400g, the latter also has a front element conventional enough to take filters and is hundreds of pounds cheaper. But check that the APS only lens is full quality.

What are the true advantages of FF over APS?

Quality. As with all camera formats, the less enlargement that is needed between the captured image and its final rendition, the higher the quality. It is also claimed that FF sensors produce less "noise" (but I dont think this applies to sensors of the same density, but it would apply to two sensors both of 10 Megapixels, spreading out the receptors reduces interference). If, historically, you were considering medium format you might want to consider FF.

Wide angle issues

Since the arrival of APS, manufacturers have made available wide angle zoom lenses of previously unknown capabilities. We can now buy wide angle zooms as wide as 8mm for APS  (same as a 12mm on a FF). These lenses, in my opinion destroy the greatest objection to the APS sensor. (see below)
The wide angle issue will of course matter if you wish to continue with your old FF lenses from film days, although adding a single lens at the wide angle end is likely to be as cost effective as a FF body.

The future

The market might move three possible ways.
a) stays the same. Maybe APS cameras concentrate on small size and FF's concentrate on ultimate quality and features.
b) sensors get cheaper and nearly all cameras end up FF, as happened with film APS/35mm cameras.
c) FF becomes a niche market or dies away.

I originally had a set of FF Sigma lenses in Pentax fit. They work fine with my K7 (APS) but are bulkier than they need be for that camera body. I have replaced them with a set of APS "digital only" lenses which are much smaller and I have some Pentax APS only lenses which are smaller still to use when size is important (mountaineering for instance). I might even consider one of the new "m" (mini) series bodies for the latter to capitalise on space saving.
Pentax seem to have jumped for the APS + "Medium" format (645D) approach, will no plans to produce a FF camera in the middle. I wonder if they are right?

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 The Sigma 8-16mm wide angle zoom, currently the widest available (2010). The only downside of this lens is the lack of filters and slightly slow maximum aperture. For me it destroys the case against APS sensors.