This is simply not a question, unfortunately, that can be answered without the assistance of a lawyer who can evaluate the scope of rights protected by the U.S. trademark, which European countries are involved, and the precise nature of the goods, services and channels of trade.
Having said that, and without knowing anything besides what you have said, you probably risk a lot by proceeding with use of the mark in sales channels, such as the Internet, that could implicate the U.S. market -- without even knowing whether the U.S. registration implicates and formal European rights or whether there are other, related, risks.
You must make the investment in consultation with an experienced U.S. commercial and trademark attorney who has the sophistication to advise you regarding this combination of legal and factual issues.
Trademark rights are specific to the countries (and, in the U.S., only those parts of the country) where the mark is used.
The owner of a U.S. trademark, therefore, cannot assert its U.S.-created rights outside the U.S. -- unless it also uses the mark in the foreign country where it seeks to enforce its rights. Because websites can be viewed throughout the world, simply publishing a web page that shows a trademark is not a use of the trademark throughout the world. Either actual sales of the product branded with the mark and/or owning a registration of the mark in a foreign country is required before the U.S. trademark owner can assert its rights outside the U.S. -- the specific rule depends on the laws of the particular foreign country.
You need to consult with a trademark attorney who practices in the country where you want to use the mark. European trademark practice is complicated because there are two different systems in place -- a "community wide" registration and registrations issued by each country.
Visit http://bit.ly/2dpgmD to search for marks that are registered and applied-for in the European Union.
Congratulations on finding the other company's use of its mark & name before you invested a lot of time and money in your mark.
As an initial matter, the "completely different logos and stylings" you mention may not be as important as the underlying words. For instance, someone trying to use "COCA COLA" in purple block letter font would still likely receive a cease & desist letter from a certain company in Atlanta, Georgia.
Turning back to your question, ironically it is simultaneously too specific and too broad to be answered fully in this general discussion forum.
You will want a trademark attorney to investigate and confirm that the company trading in the U.S. doesn't have the same or similar mark registered or pending in the European Union (or individual member states), for instance, through an international trademark filing system known as "Madrid Protocol" or via "Community Trademark" (CTM). Further, the Paris Convention says that "any person who has filed a trademark application in one of the signatory countries possesses a right to claim that filing date for priority purposes for trademark applications filed within six months in other signatory countries."
Even if specific facts of your situation suggest that you might get away with using the mark in Europe, your question seems to admit that you will be the junior user of the mark; i.e., the newcomer. Thus, even if you intend to only trade in Europe under the mark and the other company is only trading in the U.S. for now, the channels of trade may intertwine at some juncture and cause you, the junior user, problems down the road. If the other company has the ability, inclination and legal position to take you to court somewhere, it could be quite a headache and expensive.
Given that you are aware of the other company's use, it might be easiest to steer clear of a questionable mark at the outset; otherwise, in addition to legal expenses, you could be forced at some point to change your mark and therefore, your equipment, webpage, business cards, letterhead and so on ----> $$$$. Also, you may need to disclose your knowledge of the other company's prior use to a trademark examiner, or the examiner will find the senior use anyway, and/or the other company could oppose your registration of the mark. In any of these scenarios, you could end up spending money on a mark that you are unable to register or even use under "common law."
Only a trademark attorney can, after reviewing all the facts of your situation (including the mark itself, its underlying goods/services, and the other company's registration), advise you whether you should use the mark you have in mind.
In the meantime, You can find basic information about trademarks here:
and here: www.uspto.gov.
In addition to the excellent comments posted previously by my colleagues, I'd mention that it's possible that a trademark attorney might advise you to consider contacting the company with the U.S. trademark registration and seek a "live and let live" agreement. That is, if you are eager to adopt the business name that you contemplate, it's possible that you could follow the adage "good fences make good neighbors" and start with an agreement with the American company that defines how each of you would use the trademark in the respective markets in a way that would avoid confusion by your respective customers.