To solve the given optimization problem, we first need to convert the natural language description into a symbolic representation. Let's define:

- $x_1$ as the number of glass jars packaged
- $x_2$ as the number of plates packaged

The objective function aims to maximize profit. Given that the profit per glass jar is $2 and per plate is $2.50, we can represent the objective function as:

$$\text{Maximize: } 2x_1 + 2.5x_2$$

The constraints are based on the available resources:
- Worker time: $15x_1 + 12x_2 \leq 620$
- Cardboard: $3x_1 + 4x_2 \leq 120$
- Non-negativity constraints: $x_1 \geq 0, x_2 \geq 0$ (since we cannot package a negative number of items)

Thus, the symbolic representation of the problem is:
```json
{
    'sym_variables': [('x1', 'glass jars'), ('x2', 'plates')],
    'objective_function': 'Maximize: 2*x1 + 2.5*x2',
    'constraints': ['15*x1 + 12*x2 <= 620', '3*x1 + 4*x2 <= 120', 'x1 >= 0', 'x2 >= 0']
}
```

To solve this linear programming problem using Gurobi in Python, we can use the following code:
```python
from gurobipy import *

# Create a new model
m = Model("Packaging_Problem")

# Add variables
x1 = m.addVar(name="glass_jars", vtype=GRB.CONTINUOUS, lb=0)
x2 = m.addVar(name="plates", vtype=GRB.CONTINUOUS, lb=0)

# Set the objective function
m.setObjective(2*x1 + 2.5*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(15*x1 + 12*x2 <= 620, name="worker_time")
m.addConstr(3*x1 + 4*x2 <= 120, name="cardboard")

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print(f"Optimal solution: {x1.varName} = {x1.x}, {x2.varName} = {x2.x}")
    print(f"Maximum profit: {m.objVal}")
else:
    print("No optimal solution found")
```