I will discuss the both of them and how to train for each.
The Difference Between Muscle Hypertrophy and Strength
Muscle Hypertrophy
Muscle hypertrophy, otherwise known as muscle synthesis, is a process where the muscle gets bigger, mainly through sarcoplasmic hypertrophy. Although there is also myofibrillar hypertrophy (Also known as sarcomere hypertrophy) which is a bit different from sarcoplasmic hypertrophy. Hypertrophy relies on nutrition a lot more than strength does in the long run. Muscle can't be built with nothing.
Muscle hypertrophy is started off when a muscle is damaged after a workout. There are certain hormones that effect this process such as testosterone, growth hormone, insulin-like growth factor and insulin. When the muscles are repairing a few things happen. First the muscles swell up. This causes the blood to circulate through them. This means that more nutrients and oxygen are delivered to help repair the damaged muscles.
Strength
Strength is a different type of gain. It's a process where the muscles get used to being activated. The body gets used to muscles firing via neural adaptions, which means that to get stronger you need to rely less on hypertrophy and more on neural adaptions. Strength is also a bit easier to improve, especially in the short term. Some studies showed that people that are new to training could lift up to double the original weight used in a matter of weeks via a phenomenon called "noob gains". Another fun fact is that the muscle gets a bit denser via myofibrillar hypertrophy which unlike sarcoplasmic hypertrophy helps with performance.
To be more exact, this is how strength is improved: when you train, the body "remembers" the movement patterns for each exercise making you more efficient at activating the muscle groups used during that specific exercise.
There are however a few well known supplements that can help increase performance short term. A few of those supplements are citrulline-malate, beta-alinine and caffeine.
How to Train
Now that we have everything covered, lets talk about the most interesting part on this topic - how do we train for each of these? The answer is simple, more reps with less weight for muscle hypertrophy and less reps with more weight for strength. I would recommend 15 to 20 reps for muscle hypertrophy and 1 to 5 reps for strength.
Some research states however, that for strength gains it is optimal to do 1 set per exercise, while for muscle hypertrophy it is generally recommended to go high on the volume and do about 4 sets per exercise, so further into the details, I would recommend the following:
Muscle hypertrophy
Reps: 15-20 (Go higher if you are a beginner and lower if you are intermediate/advanced)
Sets: 3-5 (Go lower if you are a beginner and higher if you are intermediate/advanced)
Exercises: 5-7 (Go lower if you are a beginner and higher if you are intermediate/advanced)
Rest: 30-120 seconds (Go higher if you are a beginner and lower if you are intermediate/advanced)
General tip: choose a few compound exercises for a bigger growth hormone release and to get the muscles ready to work and then a few isolation movements to cool down at second half of the workout, to get the blood flowing into the muscle (Via the phenomenon called "The pump" and generally through contraction with high reps). You should also consider adding slow eccentrics.
Who is this optimal for: this approach is optimal for bodybuilders and teenagers looking for an aesthetic look. It is a very good approach for teenagers as the low weight training reduces the risk of injury.
Strength
Reps: 1-5 (Go higher if you are a beginner and lower if you are intermediate/advanced)
Sets: 1-3 (Go lower if you are a beginner and higher if you are intermediate/advanced)
Exercises: 3-5 (Go lower if you are a beginner and higher if you are intermediate/advanced)
Rest: 90-180 seconds (Go lower if you are a beginner and higher if you are intermediate/advanced)
General tip: try to get rid of isolation exercises and aim for compound exercises. I must also add that quick and explosive concentric movement is the best way to go for strength and power.
Who is this optimal for: this approach is optimal for people trying to compete in strongman, powerlifting, playing sports or who do manual labor. This type of training will promote muscle activation efficiency which in turn might mean saving a bit of energy doing things like manual labor or sports.
Conclusion
As we can see, they are both different and are both trained differently, and fun part is, we can combine them to get the best out of both worlds. To combine them you can simply just follow a split such as the push/pull/legs split, first three days you would do strength training then the next three days you would do hypertrophy training. With this type of training you can be a power-builder, a mix between a bodybuilder and a power-lifter.
NOTE: When I say a specific amount of reps, I do not mean picking up pencil weights and just flinging that weight for the specified reps, but I mean that you should use a weight at which you would struggle to do the needed amount of reps.
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