Chronology is an important aspect in this case. Let's first understand how Beryllium was discovered.
Emeralds and Beryl were known to ancient civilizations, they were first recognized as the same mineral ($\mathrm{Be_3 Al_2(SiO_3)_6}$) by Abbé Haüy in 1798. Later that year, Louis-Nicholas Vauquelin, a French chemist, discovered that an unknown element was present in emeralds and beryl. Attempts to isolate the new element finally succeeded in 1828 when two chemists, Friedrich Wölhler of Germany and A. Bussy of France, independently produced beryllium by reducing beryllium chloride ($\mathrm{BeCl_2}$) with potassium in a platinum crucible.
Since the salts of beryllium had a sweet taste, the element was named "glucinium" from the Greek "glykys" for "sweet" and was symbolized as "$\mathrm{Gl}$". Bussy preferred this name but Wölhler was unhappy and preferred the name "beryllium" from the Greek word 'beryllos', which means the mineral beryl. Martin Klaproth had already pointed out in 1801 that yttria also forms sweet salts. A name derived from 'beryllos' would be less likely to cause confusion than one derived from 'glykis.' Klaproth also noted that a genus of plants was already called "glucine". Finally, in 1949, IUPAC chose beryllium as the element’s name and this decision became official in 1957.
By the time the change was accepted, boron was already discovered in 1808 by French chemists Louis-Joseph Gay-Lussac and Louis-Jacques Thenard. The name came from the Arabic "burqa" for "white". They symbolized the element as "$\mathrm{B}$". Since "$\mathrm{B}$" was already taken, IUPAC had to symbolize beryllium as "$\mathrm{Be}$".
References
- History of the Origin of the Chemical Elements and Their Discoverers by N. E. Holden (PDF)
- Beryllium History - It's Elemental
- Discovery of Beryllium by Dr. Doug Stewart (Link)