## Step 1: Define the symbolic representation of the problem
Let's denote the number of dinning tables as $x_1$ and the number of coffee tables as $x_2$. The profit on each dinning table is $200, and on each coffee table is $100. The cost to make each dinning table is $250, and each coffee table is $150. The total monthly demand for both types of tables is at most 200 units, and the monthly manufacturing budget is $20,000.

## Step 2: Formulate the objective function
The objective function to maximize profit is: $200x_1 + 100x_2 - 250x_1 - 150x_2$. Simplifying this gives: $-50x_1 - 50x_2 + 200x_1 + 100x_2 = 150x_1 + 50x_2$.

## 3: Define the constraints
1. Total monthly demand constraint: $x_1 + x_2 \leq 200$
2. Monthly manufacturing budget constraint: $250x_1 + 150x_2 \leq 20000$
3. Non-negativity constraints: $x_1 \geq 0, x_2 \geq 0$

## 4: Symbolic representation in the required format
```json
{
'sym_variables': [('x1', 'dining tables'), ('x2', 'coffee tables')],
'objective_function': '150*x1 + 50*x2',
'constraints': [
    'x1 + x2 <= 200',
    '250*x1 + 150*x2 <= 20000',
    'x1 >= 0',
    'x2 >= 0'
]
}
```

## 5: Convert the problem into Gurobi code
```python
import gurobi

def solve_optimization_problem():
    # Create a new model
    model = gurobi.Model()

    # Define the variables
    x1 = model.addVar(name="dining_tables", lb=0, ub=gurobi.GRB.INFINITY, obj=150)
    x2 = model.addVar(name="coffee_tables", lb=0, ub=gurobi.GRB.INFINITY, obj=50)

    # Add constraints
    model.addConstr(x1 + x2 <= 200, name="demand_constraint")
    model.addConstr(250*x1 + 150*x2 <= 20000, name="budget_constraint")

    # Set the model to maximize
    model.setObjective(x1.obj * x1 + x2.obj * x2, gurobi.GRB.MAXIMIZE)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Dining tables: {x1.x}")
        print(f"Coffee tables: {x2.x}")
        print(f"Max Profit: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```

The above python code seems mostly correct but there was an error in defining objective function for the model, here is corrected code:

```python
import gurobi

def solve_optimization_problem():
    # Create a new model
    model = gurobi.Model()

    # Define the variables
    x1 = model.addVar(name="dining_tables", lb=0, ub=gurobi.GRB.INFINITY, obj=150)
    x2 = model.addVar(name="coffee_tables", lb=0, ub=gurobi.GRB.INFINITY, obj=50)

    # Add constraints
    model.addConstr(x1 + x2 <= 200, name="demand_constraint")
    model.addConstr(250*x1 + 150*x2 <= 20000, name="budget_constraint")

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Dining tables: {x1.x}")
        print(f"Coffee tables: {x2.x}")
        print(f"Max Profit: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```