I started training some months ago (mainly strength), but when I do sit-ups/ crunches I tend to forget counting repetitions and just continue until I can't do any more.
What are the pros and cons of counting when training?
Different rep schemes produce different effects on the body. If you use enough resistance so that you can only do 3-5 reps, you'll primarily develop strength and power. If you use a lot less resistance so you can do 30 reps, you are primarily developing muscular endurance, pain tolerance, etc.
Moreover, you should track how many reps you can do and what resistance you used so you can track progress. Increased strength/size/endurance/etc is the body's adaptation to stress. However, the body adapts exactly and specifically to the stimulus that stresses it. This means that once your body adapts to handle, say, 10 reps at some weight, if you keep using the same weight/reps over and over, your body will NOT adapt any further. In other words, if you don't progressively increase the difficulty, you are not increasing your fitness and are just spinning your wheels (well, maintaining your fitness, at best).
It gives you a basis for comparing yourself to yourself. If you could only do 10 pushups for years, then tried a new exercise and all the sudden you can do 50 pushups, you might want to example that and see why there was a change.
For weight lifting, you don't want to do too many reps. For example, don't more than 20 leg extensions means you're probably not moving enough weight. You need to balance out your weight and reps to keep them within certain ranges.
Besides, without counting how will you ever be able to brag to your buddies?