suppose we contact everybody out there who uses any one of these two apps professionally and ask what are they using them for. What would be common answers?
Is the set of users of Acrobat different from set of users of Photoshop or are they closely related?
Are there typical job designations for people using these apps professionally?
I assume you’re talking about Acrobat Professional (ability to create PDF files).
In my experience, there isn’t much overlap between Photoshop and Acrobat Pro users. Yes, many Photoshop owners also have licenses for Acrobat (because it’s included with the Adobe software collections) but that’s not the same thing as actually using Acrobat every day.
There are Acrobat users outside of graphics design. Examples include government and business office workers creating PDF forms. Another example includes folks that create marketing materials in MS Word or Powerpoint and save them as PDF for dissemination.
Check out the following Monster results. See if they provide any insight:
Photoshop results:
Monster Jobs - Job Search, Career Advice & Hiring Resources | Monster.com | Adobe Photoshop
Acrobat results:
Monster Jobs - Job Search, Career Advice & Hiring Resources | Monster.com | Adobe Acrobat
Acrobat is pretty useful for lawyers (and probably anyone else who deals with documents). Here’s a blog adobe publishes that’s aimed to that particular market: http://blogs.adobe.com/acrolaw/
Photoshop is less useful for the (ethical) lawyer 
The most common users of Photoshop would be photographers, of course, and designers. Photoshop is by far the most common photographic post-production application in the professional world. I would say the vast majority of photos you see in any newspaper or magazine has gone through Photoshop at some point.
Acrobat is also a useful engine for generating laboratory reports. Enter test data into a database, use a call (?) to gather test data, customer information, boilerplate, etc., and generate the whole thing as a PDF. This can be done without Acrobat, but Acrobat will let you go back and modify the report, add pages, etc., and lock it up so it can’t be altered by anyone else.
I’m a technical writer. My colleagues and I use both. Admittedly, Photoshop is overkill for the sort of diagrams I do, but Acrobat is essential for creating complex documentation in PDF format.
I’m a geologist and I make large and complicated graphical files of oil wells. I use Acrobat to turn them into simple files that other people can use.
In publishing, Acrobat’s markup tools are widely used for electronic proofreading. No more hard-copy proofs. It’s all on screen, just as editing is moving from paper mss. to Word (using Track Changes) and other programs.
adobe Photoshop and Adobe After Effects are also used together.
Photoshop is better to create graphics that you want to use in a video.
After Effects can read a Photoshop psd file. If I recall correctly you can use specific layers, and even adjust opacity of the layers in After Effects.
I use Photoshop to touch up pictures – simple stuff like cropping, fixing the light/dark, etc to more complicated stuff like softening wrinkles, fixing closed eyes, etc.
I don’t use Adobe Effects at all.
The markets for Photoshop and Acrobat are so different, Adobe even reports them under different divisions in their financial statements. Acrobat is part of the “Knowledge Worker” division, whereas Photoshop is part of the creative division (sorry, I forget Adobe’s actual name for that). If you dig through their SEC filings, you can even get some interesting numbers for how the divisions do in terms of sales.
To give you an anecdotal data point, I use Acrobat Pro extensively for document management. I’m a CPA and we scan everything in the office that we need to keep. In addition, it’s often useful to scan a form and then use comment/markup tools to type in information rather than handwriting anything. I even modified a state sales tax form to use JavaScript calculations on form fields. So we literally use Acrobat every day.
I have an old version of Photoshop and use it for graphics manipulation such as for business cards and ads. That is needed maybe five or six times per year.
I build my cartoons in Photoshop (and also Illustrator sometimes).
I use Acrobat as a final pre-press proofing stage for books, after creating them in other applications (Framemaker in my old job; InDesign now). The PDF goes around for review, then it gets sent to the printer’s.
A lot of logos and such are done in Illustrator or another vector application; this allows the designs to be resolution-independant. They can be scaled to whatever size is needed before being rasterized (converted to pixels) and printed.
I have seen so many logos that were scrunched down and then rasterized to fit in a small space… and then blown up again, and the pixels just get bigger. Information has been lost, and there’s no way to restore the original smoothness. It’s like pulling teeth sometimes to get the original vector file.
Raster-based programs like Photoshop are good for things like photographs and scans of original artwork. Vector-based programs like Illustrator are good for line-based artwork such as drawings and logos. There are ways in both Photoshop and Illustrator to combine them, but each program handles one type of artwork best.
I’m an artist and designer and photographer, and I use Photoshop several hours every single day of my life. I have Acrobat, but rarely use it.
I own a small newspaper. I use Photoshop constantly; ad design, cover design, photo adjustments (mostly color, special effects, and levels – I don’t “enhance” people in the pictures).
I use Acrobat for proofs, and to send the final “camera-ready” copy from InDesign to the print shop. A few customers send me ads in PDF format. That’s it.
Is Acrobat even needed for that? I’m not doubting you – I’m genuinely curious about what Acrobat adds in those use cases, because I might be missing out on something useful 
I’ve been using Adobe products (primarily Photoshop, Illustrator, InDesign and PageMaker prior to Indesign) since around 1992, and I’ve used Acrobat Pro maybe 10 times (I do have a license as part of Creative Suite Design Standard). For outputting PDF for output or to send to people, InDesign and Illustrator do that fine on their own (for my needs, at least). For reading PDF, Preview or Acrobat Reader are plenty. The primary use I’ve made of Acrobat Pro is combining PDFs and encrypting/password protecting them (typically for personal stuff unrelated to design work). There are probably lighter-weight ways to do both of those things, but Acrobat is already there and paid for, so I use it.
Acrobat Pro has a lot of powerful preflight and PDF cleanup tools that are very useful if you you are distibuting files to print manufacturing using one of the ISO PDF/X standards. And even though exporting a PDF out of InDesign or Illustrator works most of the time, there are still today times when processing a complex file that only dumping a PostScript file and running it through Distiller will work.
And the markup, commenting and regression tools in Acrobat Pro are super useful in a busy creative department too.
I’m a book editor. Books come to me in (usually) Word or WordPerfect (WordPerfect: still used by law librarians all over the country). I convert it to something else, often InDesign, and sometimes I have to convert again, using Adobe, into a PDF format to send out for proofs, or to the printer.
PhotoShop, I use, but all I have and all I need is PhotoShop elements. I use it to size and crop pictures, convert them to B&W. and do them as halftones so they print nicely in B&W. It can be handy to convert photos people send in formats other than JPG, too. In a couple of instances, usually for covers, I’ve used it for 4-color separation, but InDesign does that, too (and does it better IMO).
ETA: Basically, what the Wombat said.
I’m an illustrator and graphic designer and I use Photoshop to digitally paint, adjust color levels and resize. I would say I use Illustrator and Photoshop around equally. I use Acrobat pretty regularly to view files. I generally use InDesign or Illustrator to generate my PDFs so I can get proofs for my clients.
When I send stuff out to print it’s always as a PDF nowadays. Before the PDF workflow you used to have to collect files and upload a native document, links and fonts to get stuff printed. The print universe is much happier since PDFs came around.
I’d like to second that any professionally made logo should be done in Illustrator or similar vector drawing program.
Yes. We used to run our postscript files from FrameMaker through Distiller to get the final PDFs for 300-page manuals. But even generating the PDFs directly in an application or in Acrobat meant we avoided having to deal with a different printer driver for each printer and all the variables that entailed.
The ‘preflight’ tools in Acrobat do things like listing what drawing objects are in spot colours and which are in process colours. This is important if your job is supposed to be set up on an offset press with only certain spot colours, say a specific shade of yellow ink and a specific black, but you are importing artwork from various sources.
If the imported artwork was made in CMYK, it will go into the PDF without problem, but will trigger the creation of a set of four CMYK plates when separations are made. These plates are in addition to the two spot-colour plates you were intending. Then, when the printers get the PDF, they’ll find that it calls for six plates instead of two to put on their presses. At this point they would call me back and ask what was going on…
I learned to preflight my PDFs very carefully for spot-colour work. Yet in many ways spot-colour work is easier than four-colour process work. Because spot colours are externally specified, for instance by a Pantone colour reference, all they needed to be was labeled and identified in the PDF.
There was no need for colour calibration so that the monitor and the printer both looked as similar as they could. So what if the colour looked orange on the test print from the local laser printer, and lemon on the monitor? It was still Pantone 109 C, and the printer would have the real colour of ink on hand.