Edinburgh must win some silverware before they can consider themselves of the same stature as their rivals, says head coach Richard Cockerill.
The capital side top Pro14 Conference B, and two wins could secure a home semi-final and the chance for a first major trophy.
They face Glasgow twice as the league returns after five months, with the first match on Saturday.
"The reality is that we've done well for us," Cockerill said.
"But we haven't won any trophies and we haven't got any historic silverware in the cabinet.
"Glasgow have won the competition. They've been in finals and semi-finals so they have that bit of pedigree that we don't have. We're building that."
Cockerill believes his side are "miles better" than when he took on the job in 2017, and much closer to success.
Having missed out on the play-offs last season, the former Leicester coach stressed the opportunity now in front of his squad.
"I'm not going to underplay it," he said. "We've got to this point so why would you give it away easily now?
"We're going to go as hard as we can over the next two weeks. If we're the least favoured [of the last four] then great. It will suit us."
Talks between Cockerill and Scottish Rugby about about a contract extension are ongoing, with the Englishman's current deal due to expire in 2021.
"They have every intention to want me to stay here and I have every intention to stay, so I think it looks pretty positive," he added.
"Hopefully in the next few weeks we can sort something out and start building our squad to the next World Cup and beyond."