The biggest advantage you can gain is if you file two identical trademark applications within 6 months of each other in different countries. This would grant you something called 'priority filing'.
With priority filing, the subsequent trademark application filed in a different country can use the earlier filing date from the prior trademark application. What this means, in practice, is that if you file your EU trademark application on the 1st of March and your US trademark application on the 1st of July, the US trademark application can use priority filing to gain the filing date of your EU trademark application, which would be the 1st of March instead of 1st of July. An earlier filing date means your protection would have begun sooner, which is an attractive perk to gain.
In the US, it is also possible to claim commercial use (an important and mandatory part of the process in the US specifically) based on a trademark registered outside of the US. This way, you can skip the submission of the Statement of Use, a statement that your trademark is used in commerce.
However, besides these benefits, having a pending or even a registered trademark in one country won't affect your chances when applying in another, as your application will be assessed independently by the country's IP Office.