There is a large number of different drugs in the group known as non-steroidal anti-inflammatories. The group includes ibuprofen (Neurofen), diclofenac (voltarol) and meloxicam (mobic). ALL of them can cause gastro-intestinal side effects including bleeding and ulceration.
Some drugs are marketed as being less likely to cause side effects but this is usually manufacturer hype. The likelihood of a NSAID causing GI side-effects is probably linked to its potency i.e. the strongest anti-inflammatories are more likely to cause stomach problems. This is why ibuprofen can be sold over the counter because it is a relatively weak anti-inflammatory with the least chance of side-effects.
The anti-inflammatories were originally developed for people with inflammatory conditions such as rheumatoid or osteoarthritis. People with these conditions experience 'inappropriate' inflammation which causes disabling symptoms.
Of course it was found that the drugs were also pretty good pain killers for people with a host of musculo-skeletal problems. There is some truth in the suggestion that inflammation is a normal physiological response to injury, however there is evidence that NSAIDs can speed healing of injuries.
All of them, including the Vet's meloxicam, should be used with caution, in as low a dose as possible, for as short a time as possible with an eye out for significant GI side-effects.