No difference in visual effects.
The difference is in how they achieve the same result (filter's internal construction, etc.), and more importantly, the difference is how they affect metering and focusing systems. Linear polarizers might interfere with modern camera's metering and focusing systems, hence the safe bet is to go with a circular version. If you use older, manual focus, manual metering cameras (say Hasselblad V-series), then a linear would do fine.
Is multi-coating worth double the price? Depends on your level of perfectionism. For most shots probably not, but if you have once-in-a-lifetime shot ruined by a reduced contrast and/or flare, then it might make sense to pay more. In addition to that, some modern multi-coated filters use a special coating that repels water, dust and fingerprints at the same time. Again, depending on your perfectionism, might be well worth the extra price.
Hope this helps.
