Quite often it's for efficiency. Let's take the Australian arrivals card as an example, as it has quite a few questions compared with some.
Sample arrivals card for Australia
Some of the questions:
- Do you intend to live in Australia for the next 12 months?
- Do you have criminal convictions?
- Are you bringing in [food/drugs/medicine/money] into the country?
Now some of these are going to be simple 'no' answers for the most part, for most passengers. This saves customs people asking the same questions over and over again. Instead, when they see a 'yes', then they can ask. This speeds up the process for all, and more importantly, lets the customs people deal with the 'special cases' - the ones that actually matter and are what they are there for.
Do you intend to live in Australia? No = tourist. Yes = time to ask about how they support themselves, where they'll be working, etc.
Are you bringing in drugs? This one I occasionally answer yes due to the wording, as I have prescription drugs. I've sometimes been questioned on it, but usually the word 'prescription is enough'.
Now you might wonder what idiot with illegal drugs would say 'yes'. Fair enough. However sometimes the person doesn't realise their medicine from Timbuktu (random example) might be illegal or controlled substances in Australia. So it's necessary to ask to clarify, and when the person says yes, they might have a doctor prescribe them something controlled as a substitute which is allowed in the country.
Furthermore, in the event of a post-customs search of their bag, there's an additional legal benefit of asking these questions - it's made all passengers legally consider the question. And when you find drugs/food/etc in their bag, they can't say 'I didn't realise I had to declare it' - they literally just signed that they didn't have it on a form.
Now not every country has this. Some just want a place of address. I always used to be annoyed by this, especially after seeing someone use the address from Pretty Woman. Until I left my luggage at the airport (took the wrong bag), and they were able to contact me on this. While chatting to them about it (the address was the one on my bag), we were talking about the other ones and the arrival one can be used in case of emergency too. If, given the current Ebola scare, for example, a passenger on your flight was found to have it, this gives them a means to attempt to contact you to check for early symptoms and limit exposure.
So yes, they're tedious, but they do still serve a variety of purposes - for record-keeping, efficiency and legal documentation.