That’s a tough question, because there is a long and complicated history between the socialist movement and the labor movement in America: at various times, the socialist movement has pursued a strategy of entryism (joining major labor unions in order to try to shift them left), dual unionism (forming explicitly socialist unions to compete with non-socialist labor unions). Likewise, at various times (the 1910s and 1930s especially) the labor movement has recruited socialist activists to help them organize workers and worked closely with left-wing organizations to promote pro-labor causes, and at other times (the 1920s and the late 40s-50s especially) ruthlessly purged socialists or communists from labor unions and barred them from membership.
Indeed, the complexity of this history can be seen in the fact that John L. Lewis of the United Mine Workers pursued both, purging left-wingers in the late 20s when he viewed them as a threat to his control of the UMW and then recruiting them in the 1930s to help him build the Congress of Industrial Organizations.
Another thing that complicates this is the decentralized and indeed fragmented nature of the American labor movement, which means that various unions (and indeed, various locals within unions) have more of a history with the socialist movement than others. Historically, for example, building trades unions and the other “skilled trades” clustered in the AFL have tended to be more politically conservative and less welcoming to the left, whereas manufacturing/”industrial” unions which grew from the CIO tended to be more left-leaning, in part because CIO unions recruited radicals to help them organize, in part because their workforces were more likely to be from immigrant and ethnic communities which had their own socialist traditions, and in part because the CIO came up during the 1930s.
However, these causal stories can get really confusing, because you have competing unions in the same industries with different political histories: the International Longshore and Warehouse Union (ILWU) on the West Coast has a long left-wing history due to their founder Harry Bridges being an open Communist, whereas the International Longeshoreman’s Association (ILA) on the East Coast was much more conservative.
To answer your last question, I don’t have a lot of first-hand experience of European labor unions, but my impression is that U.S labor unions tend to have a lot more ideological diversity – there’s unions who are proud of being kicked out of the AFL-CIO for being too Communist back in 1947, there’s unions that were more associated with Catholic social thought, there’s unions with a long history of liberal anti-communism, etc. – whereas it’s more common for European unions to have firm institutional links to a socialist or labor party. In part because of that ideological diversity and the more complicated historical relationship between the American labor movement and American political parties, American unions have a more contingent, transactional, and pragmatic political program (especially at the local level), in contrast to European unions who either founded or were founded by a political party in their country and thus have very close relationships with that party and that party alone.
With all that diversity, how is it that American unions are weak compared to Europe?
Well, I don’t think the ideological diversity is a strength or a weakness.
And in terms of strength and weakness, I think that’s driven much more by different labor law on the one hand, and very different business cultures on the other.