In common English, there is no sharp difference. One may sound better than another in a specific sentence. In technical English, there is a big difference.
In probability, statistics, mathematics and the sciences, they are very different things. A technical discussion in mathematical language can be found here.
In non-mathematical language, assume that you know the mathematical formula for some system or element of a system such as gravity for a falling object. Also, imagine that the formula allows for uncertainty. For example, the wind may blow in an unpredictable way when you are dropping an object to measure how quickly it falls. For some objects, that may not meaningfully matter, but it might for feathers.
Physical constants, such as the gravitation constant, are called parameters. If you know the relevant value of the constants, then you can calculate the probability that an object will take a value between two points. For example, you could calculate the probability that the object will take between 3 and 3.1 seconds to fall the stated distance.
When discussing a system like that, we treat any discrepancies as being due to chance. A probability, in this sense, is the long-run frequency of seeing outcomes over a range if the experiment is repeated forever.
A likelihood flips the knowns and the unknowns. Imagine that you did not know the values for the constants in your system, or maybe you knew some but not others. As long as you have seen data, then you can use the likelihood to estimate the values of the parameter.
For example, if you dropped the object and it took exactly 3.05769 seconds to hit the ground, you can calculate the likelihood that the gravitational constant takes on certain values. It is not a probability. It can be turned into a probability if you can state your beliefs about the possible values it could take as a probability. For example, if you believed there was a fifty percent the constant takes a value between 9.7 and 9.9 meters per second per second, and a twenty-five percent chance it is less than 9.7 meters per second per second, and a twenty-five percent chance it was between 9.9 and 11 meters per second per second, then you could convert that likelihood into proper statements of probability.
You do have to have beliefs that could be stated as probabilities, but if you do, then you can take the values of a likelihood and end up with probabilities.
The language in Wikipedia is technical.
In mathematics, a measure, very loosely, such as area, volume, or probability, is something that adds up to a specific value. There is some way to assign a number to the size of a set. Intuitively, you have been measuring things all of your life, but mathematics has rules as to what is a measure and what is not. A probability is always a measure that also has very specific properties.
A likelihood does not add up to a specific number. A likelihood is more like a relative statement. For example, imagine there are two explanations for some phenomenon, A and B. The likelihood of A is 1/3 and of B is 2/3. If you multiplied them by 3, then the likelihood of A is 1 and the likelihood of B is 2. What matters is that the relative proportions always stay the same. The actual value is irrelevant. If the likelihoods were expressed and 10 and 20, that would be just as valid and useful.
As a side note, when probability is treated as being due to chance because you know the parameters, then that type of probability is called aleatory probability. When you are estimating parameters by combining the likelihood of a particular observation with the probabilities of your beliefs being true, then that is called epistemic probability. Your probabilities are not a measure of chance, but a measure of your uncertainty about how the world really works.
