diff --git a/doc/source/whatsnew/v1.1.0.rst b/doc/source/whatsnew/v1.1.0.rst
index 90534c00df621..28e30e95e8bf1 100644
--- a/doc/source/whatsnew/v1.1.0.rst
+++ b/doc/source/whatsnew/v1.1.0.rst
@@ -953,6 +953,7 @@ Numeric
 - Bug in :meth:`DataFrame.diff` with ``axis=1`` returning incorrect results with mixed dtypes (:issue:`32995`)
 - Bug in :meth:`DataFrame.corr` and :meth:`DataFrame.cov` raising when handling nullable integer columns with ``pandas.NA`` (:issue:`33803`)
 - Bug in :class:`DataFrame` and :class:`Series` addition and subtraction between object-dtype objects and ``datetime64`` dtype objects (:issue:`33824`)
+- Bug in :meth:`Index.difference` incorrect results when comparing a :class:`Float64Index` and object :class:`Index` (:issue:`35217`)
 - Bug in :class:`DataFrame` reductions (e.g. ``df.min()``, ``df.max()``) with ``ExtensionArray`` dtypes (:issue:`34520`, :issue:`32651`)
 
 Conversion
diff --git a/pandas/core/indexes/numeric.py b/pandas/core/indexes/numeric.py
index 5020a25c88ff4..731907993d08f 100644
--- a/pandas/core/indexes/numeric.py
+++ b/pandas/core/indexes/numeric.py
@@ -400,28 +400,6 @@ def _format_native_types(
         )
         return formatter.get_result_as_array()
 
-    def equals(self, other) -> bool:
-        """
-        Determines if two Index objects contain the same elements.
-        """
-        if self is other:
-            return True
-
-        if not isinstance(other, Index):
-            return False
-
-        # need to compare nans locations and make sure that they are the same
-        # since nans don't compare equal this is a bit tricky
-        try:
-            if not isinstance(other, Float64Index):
-                other = self._constructor(other)
-            if not is_dtype_equal(self.dtype, other.dtype) or self.shape != other.shape:
-                return False
-            left, right = self._values, other._values
-            return ((left == right) | (self._isnan & other._isnan)).all()
-        except (TypeError, ValueError):
-            return False
-
     def __contains__(self, other: Any) -> bool:
         hash(other)
         if super().__contains__(other):
diff --git a/pandas/tests/indexes/test_numeric.py b/pandas/tests/indexes/test_numeric.py
index 33de0800658f2..a7c5734ef9b02 100644
--- a/pandas/tests/indexes/test_numeric.py
+++ b/pandas/tests/indexes/test_numeric.py
@@ -239,6 +239,19 @@ def test_equals_numeric(self):
         i2 = Float64Index([1.0, np.nan])
         assert i.equals(i2)
 
+    @pytest.mark.parametrize(
+        "other",
+        (
+            Int64Index([1, 2]),
+            Index([1.0, 2.0], dtype=object),
+            Index([1, 2], dtype=object),
+        ),
+    )
+    def test_equals_numeric_other_index_type(self, other):
+        i = Float64Index([1.0, 2.0])
+        assert i.equals(other)
+        assert other.equals(i)
+
     @pytest.mark.parametrize(
         "vals",
         [
@@ -635,3 +648,27 @@ def test_uint_index_does_not_convert_to_float64():
     tm.assert_index_equal(result.index, expected)
 
     tm.assert_equal(result, series[:3])
+
+
+def test_float64_index_equals():
+    # https://github.com/pandas-dev/pandas/issues/35217
+    float_index = pd.Index([1.0, 2, 3])
+    string_index = pd.Index(["1", "2", "3"])
+
+    result = float_index.equals(string_index)
+    assert result is False
+
+    result = string_index.equals(float_index)
+    assert result is False
+
+
+def test_float64_index_difference():
+    # https://github.com/pandas-dev/pandas/issues/35217
+    float_index = pd.Index([1.0, 2, 3])
+    string_index = pd.Index(["1", "2", "3"])
+
+    result = float_index.difference(string_index)
+    tm.assert_index_equal(result, float_index)
+
+    result = string_index.difference(float_index)
+    tm.assert_index_equal(result, string_index)