Here is what happens if you believe on what Jesus did, rather than on Him as a Person. You end up separating His offices and worshipping a 1/3 Jesus. Jesus was the Messiah, the Anointed One of God. That anointing made Him to be Prophet, Priest, and King.
If someone “trusts only in the finished work of Christ on the cross for his salvation,” he ends up accepting Jesus as High Priest—a 1/3 Jesus—to the exclusion of Prophet and King. Some people have even coined the term “saving faith,” which is not found in the Bible. While that term may not be wrong if used rightly, it is often used in the context of accepting Jesus’ work as High Priest, but not including Jesus’ work as Prophet (who proclaimed God’s new law) and King (who started a new kingdom when He came).
Can we divide Jesus up? Can we be saved if we trust in what He did, rather than who He was? Can we accept some parts of His life, but not the whole? Can we say, “I accept Jesus as my personal Savior (Priest),” and then not accept Him as our Prophet and King?
No! We cannot say, “I will drink His blood, but not eat His flesh.” John 6:53 Furthermore, we need to consider Hebrews 5:9:
And being made perfect, he became the author of eternal salvation unto all them that obey him.
Jesus became the Author of Salvation unto all that obey Him. All who disobey Him are without salvation. Jesus will certainly not be your High Priest if you do not submit to His kingship and His rules and do what He says.
And if you think you can get around that by quoting Ephesians 2:8-9 (For by grace are ye saved through faith; and that not of yourselves: it is the gift of God: Not of works, lest any man should boast.), please go read the article, which looks at the meanings of grace, saved, faith, and works, as found in the Bible.