Japanese has a complicated system for apologizing, which can change drastically depending on who you are speaking to and the level of formality you wish to express.
However, すみません and 失礼します are two of the
most versatile ways of excusing yourself or saying sorry, and can be used in a wide variety of situations without causing offense.
すみません means "I'm sorry" while 失礼します, literally translates to "I am rude."
However, on their own, the two phrases are synonymous and if you accidentally spilled your drink on the person next to you, either
one would be a suitable way of apologizing.
It is also worth noting that in Japan, people will often "apologize," when English speakers would customarily say "Thank you."
For example, if you brought someone a cup of tea, she would be just as likely to say すみません as ありがとうございます.
This might seem strange, but you can think of it as that person acknowledging the trouble you went through to do something on their behalf.