There is no actual difference in pronunciation. You might have been confused by the different ways that you saw it written (different transcriptions) in a British dictionary and in an American dictionary. However, it's not because of any difference in pronunciation. It's just because there are different phonetic alphabets, so different signs might be used in them to represent things that are the same.
As the Wikipedia article on the International Phonetic alphabet states,
Many British dictionaries, including the Oxford English Dictionary and some learner's dictionaries such as the Oxford Advanced Learner's Dictionary and the Cambridge Advanced Learner's Dictionary, now use the International Phonetic Alphabet to represent the pronunciation of words. However, most American (and some British) volumes use one of a variety of pronunciation respelling systems, intended to be more comfortable for readers of English. For example, the respelling systems in many American dictionaries (such as Merriam-Webster) use ⟨y⟩ for IPA [j] and ⟨sh⟩ for IPA [ʃ], reflecting common representations of those sounds in written English, using only letters of the English Roman alphabet and variations of them. (In IPA, [y] represents the sound of the French ⟨u⟩ (as in tu), and [sh] represents the pair of sounds in grasshopper.)
You might have just seen different ways of representing the same sound.
Please pay attention to what I've marked in bold font in the above paragraph:
use ⟨y⟩ for IPA [j]
You might just be dealing with different phonetic alphabets in your example.
The specific way phonemes are pronounced can be wildly different not only in different dialects, but simply from one person to another. In fact, there is a linguistical term for a bulk of specific sounds that are different from one another if they were to be analyzed on a spectrograph, but they represent the same phoneme. These are called allophones. In fact, a phoneme is largely an abstraction. That is why, as was noted in comments to your question, it is just best to listen to how native speakers pronounce words and to try to imitate that. (Keeping in mind that native speakers will still pronounce things slightly different, irrespective of whether they are, say, British rather than American, or vice versa.)
Bottom line is, different ways of "writing phonetics" don't necessarily mean that there is an actual difference in ideal pronunciation. There are differences in real pronunciations, because phonemes are continuums of different sounds (allophones). But you should not build a theory out of it.