This is a common question and definitely worth discussing. Tylenol (APAP) and ibuprofen (an NSAID) both are known and regularly used for their analgesic or pain reducing properties. However, both can also antipyretics and can reduce a fever. Only NSAIDs though can cause anti-inflammation and as such can reduce swelling. Taking this into consideration on their effect, we can turn to side effects. Tylenol gets referenced often to its liver toxicity, which is very true. However, one would have to consume a dose greater than what is recommended, usually (adult = 4 grams max, which may be reduced per the FDA soon). Chronic maxed dose of Tylenol may also lead to liver toxicity. But herein lies the issue, one should not be using Tylenol for self medication for > 10 days, at that point see someone. In addition, do not consume alchoholl with Tylenol as it decreases the metabolism of Tylenol, which takes place in the liver. Otherwise, Tylenol is great to use in short duration or as an add on analgesic for pain if needed and recommended. For instance, if you look at most opiods (Percocet and Vicodin), the added ingredient is Tylenol. Now NSAIDs on the other hand have their place in pain especially due to inflammation, such as arthritis or swelling due to rupture. But they have multiple side effects and are not recommended in several disease states. Aspirin sensitive asthmatics may have increased breathing problems with NSAIDS. There is an increased risk of cardiovascular events associated with NSAIDS, and patients with CHF should not be on NSAIDS as it may increase edema. NSAIDS may lead to stomach ulcers and gastrointestinal bleeds. NSAIDS, especially with some medications like warfarin, increase the risk for bleeding. NSAIDs can lead to kidney damage and failure as they are heavily excreted via the kidneys.
This is a long answer, but the drugs are multifaceted. For short pain treatment I recommend Tylenol with the instructions provided on the box. NSAIDs have their place, but have a greater population that should not utilize them. As always, for chronic pain see a healthcare provider as self medication chronically is not recommended without some assessment on the etiology of the pain.